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Un des symboles suivants apparaTtra sur la dernidre image de cheque microfiche, selon le cas: le '»ymbole —*- signifie "A SUIVRE ", le symbols V signifie "FIN". Maps, plates, charts, etc., may be filmed at different rDducticn ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to righi and top tu bottom, as many frames as required. The following diagrams illustrate the method: Les cartes, planches, tableaux, etc., peuvent Sere film6s d des taux de reduction diffdrents. Lorsque 9e document est trop grand pour dtre reproduit en un seul cHchd. 11 est fi!m6 A partir de I'angle supdrieur gauche, de gauche d droite, et de haut en bas, en prenanf: ,'a nombre d'images ndcessaire. Les cfiagrammes suivants illustrent la m6thode. 1 2 3 1 2 3 4 5 6 I \i 4 4 THE HIGH SCHOOL ALGEBRA. PART II. BY I. J. BIRCHARD, M.A, Ph.D., MtUhematical Master, Collegiate Institute, Brantford, AND W. J. ROBERTSON, B.A., LL.B., Maihemtitical Master, Collegiate Institute, St. Catharines. TORONTO. VV^IIvIvIAlVI BRIGOS. 1889. Clf)15X Entered according: to the Act of the Parliament of Canada, in the year one thousand eight hundred and eighty-nine, by Wilmam Brioos, Book Steward of the Methcdiot Book and Publishing House, at the Department of Agriculture. PBEFACE. The favomble reception accorded to "The High School Algebra. Part I by the Mathematical Masters of the leadin.. CouLiate Inst.tute, and High School, of Ontario, has induced'tha auC to proceed wth Part 1I„ which is now given to the public Ite eadmgf«.turesaresi„,ilartotho.eoftLfornK.rvoirme Pa tt tL ^ ffi u . • . ™ ^"^ «'™» considerable prominence The d,ftc„lt.es of the subject are presented one at a tireTn logical order preceded, where experience has shown it nece^slrv by numencal .Uustrations to prepare the way for more gene mi mvesfff»t.ons Explanatory matter and fori] proofs of 11 s.tions have been kept distinct, as far as possible, fo'theX vemence of students preparing for writte/examiLtions The more important theorems, which should be read by all student, readers and those who are not candidates for Honors o2 „ ahty has not been attempted; yet new views of old th" rem wm be found ,n many instances, and new theo,^ms, also, ira7ew :::i^^;r ^--^ '" ^"' ---"^ -^--^ '~ The exan.ples, which are very numerous and varied in their character, have all been tested in the class room, and proved "o be suitable before being inserted. Their number is greater than the majority of students will find time for working- but !^ . are carefully craded it will k. . , v^orjjing, but as they desirerl nf T ^^'^ **" '^^^'^* ^ "^^^7 ^s may be desired of any required degree of difficulty. During the fir^t two-thirds of the examples will be amply sufficient. Aa effort has been made, by „,eans of diagrams and familiar I IV PREFACE. If illustrations, to show clearly the connection between the symbols on paper and the actual quantities they represent. This will be especially noticed in the chapter on Imaginary Quantities, which has been treated wholly from a geometrical point of view. Ex- perience seems to warrant the belief that this method will prove interesting and instructive to the student who limits his atten- tion to ordinary Algebra; whilst to those who pursue their way through the higher mathematics it will serve as an introduction to the new and beautiful science of Quaternions. Throughout the whole work the authors have constantly kept in mind the future as well as the immediate wants of the student. The treatment of Homogeneous Equations will be found con- venient in Conic Sections; the Theory of Infinite Quantities pre- pares the way for the Calculus; whilst it will afterwards be seen that many of the examples give the solution of problems in various departments o^ more advanced work. The materials used in the preparation of the present work have been gathered from many sources. The standard Algebras of Wood, Potts and Todhunter have furnished a considerable por- tion. The more recent works of Chrystal, C. Smith, Whitworth, Hall + c)m = am~bm-\-cm; {a-b){c-d)^{a-b)c~{a-b)d = a^-bc-ad+hd; (a + b-c)~-m = a^m + b-^m-c~m. 4. The exact meaning of tlie examples in the preceding Art should be carefully noted, expressed in words, and, where possible! Illustrated by concrete quantities. Thus in the third examme under the Law of Distribution the first combination of symbols directs us to subtract b from a, add c to the difference, and n ulti^ plv the Slim h^T 1, a<«==^. awf^^7«< l^^oc^o I*t «=|, then, by taking the 2nd, 3:d, 4th, etc., powe« in succession, it wiU be observed that e.ch multiplication adds mo. than - to the original fraction ; therefore bv ™.,u._,„-_ _ „ '-• •"'^^i^ijiiig c4 sum- FUNDAMENTAL CONCEPTIONS AND OPERATlONa 15 cient number of times the result way be made greater than any finite quantity, or a* — oc. Again, if a < 1, let a= - ; thenp > 1 and ««* = _L„i_=:() oc The truth of this latter proposition is generally assumed by merely noting that if a is less than unity, each term of the series a, a\ a\ etc., is less than the preceding, and therefore by taking the exponent large enough the result may be made less than any finite quanoity. But this mode of reasoning is fallacious, for the terms ofthe series -, yj, (-j, (^j, etc., continually decrease, and yet if the series be continued to any extent the terms will always be greater than -. o 16. Fractions which take the form - when particular values are given to the literal symbols involved are termed Vanishing Fractions. They usually arise from the numerator and denomi- nator having a common factor, which is zero for the given values. Such fractions have no definite value if by we mean the entire absence of quantity; but with the meaning assigned in Art. 10 a definite value may generally be found. Ex. i.— To find the limit of the value of — ^ when the value x~a of X approaches the value of a. let the value of h become indefinitely small, then the value of the fraction, viz., 2a + /i, becomes indefinitely near to 2a, i.e., by making x sufiiciently near in value to a, the value of the fraction may be made as nearly equal as we please to 2a, the limit required. Pranfinallv fViJa tu^a.-iH i^ e i _i i _. „ .vcT.tiv io Avuiiu ill, uuco oy removing the com- uion factor x-a and writing a for x in the quotient. u fliOHfiR AteaBftA. ^^^ ^.-To find th^ value of '^±^ ^fc^n .= -1 a„d whenar=.oc. When ar= - 1 both numerator and denominator vanish, there- fore :r + 1 is a factor. Removing thJ. factor we get ^^ ; sub- stituting - 1 for X ve get - 1, the result required. ^""^^ ' When X is infinitely great both terms of the fraction become oc; Its value, therefore, in this form is indeterminate. The frac- tion, however, in its lowest terms may be written t; and If we now put ar=oc, - and t each =0, and the fraction be- comes - . The meatxing in this case is, « By making x sufficiently great the value of th^ fra<5tion may be made as nearly equal as we please to - ." o 17. The product of two factors vanishes when one factor van- ishes, providing the other remains finite; if the second factor be- comes infimte the product may be zero, finite or infinite, as the tollowing simple examples show: Leta? = a2 andlet(l)«=l (2)v-- {'\\,. ^ . +u • , " \ I y ^y \^^) y-^2» \^) y^—zy *hen in each case when ar = 0, a = Oandy=oc Thus, 1 (1) xy = d?,-^a^(i. n. a a? (3) .ry = a2.-=_=,oc. a? a In the above and all simUar examples the meaning assigned to iT^lt In"' "'"" *''° "'^'""'■y "^™S the abovfprocess would be wholly unintelligible. It should be further oWved the .wo .actors « «,„e, the one 0, the other =c, by the vanishing FUNDAMENTAL CONCEPTIONS AND OPERATIONS. 17 of the same quantity, a; otherwise no definite result could be given. For example, if ar = a and ?/= 7, and if u and b each be- 1 * come 0, then ary = a x -= x oc ; but this product is entirely in- definite. EXERCISE ', 1. Find the value of integer. x-a when r = ffl, w being a positive 2. Find ^he value of when ar= -a, (1) if n be odd; (2) if n be even. 3. Find the value of x + a 1 - 3.t2 + 2^5 (1 - xy - when x=l. 4. Find the value of - — — - — — when a- = and when or = oc. ar + 3.r - 4.r 5. If a?= 1, find the value of — — — - when n= 1, 2 and 3. (a? - 1 )" 6. Find the value of K-^) +(a-x) ^^^^ ^^^^ T -p- J xi, 1 ' e ^^- V2«+ V'x - '2a , / . -b md the value of when x = 2a. Vx^ - 4tt2 8. Find the value of — ^^ — when a = 0. 9. Find the value of a 2a when ar = and when a; = oc. I -a' 10. Find the value of — — - when ar = « ard ?/= i. 11. If a:' + y2-(2y + a-% + (a-% = ai, find the value of ?/'-{a^ + 2h^)i/ + 2ab^ when ?/ = a. , CHAPTEE II. If ! : RATIO. magnitudes only bv means of f k/ u ^^^""^ ^^^^^ "^'^^ and EucHd. deUrrntt^i^Srel"!?''^^ are therefore comnellp^ f^ • ^P^^^P"^*® ^^^ *hat purpose; we though i„^«a,;sf. w e^l; f^r "f""™' *'='■• us to W the subject numeri^^. ' *'" ™"='' "^ *"' •'"-Wo w;o:it.^X:rr^L: r r,r -r - ■ the unit. ^ ^"^ ^^^^i" is taken as «.e .tic 05 3 feet^^'^H-hJ ITrer l[ T ,/r"^^'^ fuUy observed that ratb exi... In't'C!^'- '"'"''' '^ """^ ""v '-^v-.v-oua qua/iUtties of the BATIO. 10 same kind, for the unit of measurement must evidently be of the same nature as the quantity to be measured. 22. Since the values of ratios are measured by fractions we are enabled at once to add, subtract, multiply and divide ratios by the rules which govern these operations in fractions, and all theorems which have been proved for fractions are equally true for ratios. For example, The terms of a ratio may be multiplied or divided by the same number without changing its value. In this connection see Arts. 151 and 169-171 of Part I. 23. A ratio is said to bo "a ratio of greater inequality," «'a ratio of equality," or " a ratio of less inequality," according as the antecedent is greater than, equal to, or less than the consequent. In connection with this definition only the numerical values of the antecedent and consequent are to be considered, otherwise it would be inconsistent with previous definitions. For example, 3 : 4 is a ratio of less inequality, since 3 is less than 4 ; but if this restriction were removed, - 3 : - 4 would be a ratio of greater inequality, since - 3 is algebraically greater than - 4. But the 3 _ ^ ^ value of the former ratio is - and that of the latter — -= -, i.e., a ratio of greater inequality would be equal to a ratio of less in- equality, which is absurd. In the Theorems which follow the terms of the various ratios Te considered positive. 24. A ratio of greater inequality is diminis/ted, and a ratio of less inequality is increased, by adding the same positive quantity to both its term^. Let a : J be the original ratio, and let a + ar : i + a? be the ratio formed by adding the same positive quantity to both its terms. a a + x x(a - b) Then and this result is positive or negative as a is greater or less than b. 20 HIGHER ALOEBHA, Therefore, if a > i, and if a < A, which proves the proposition. b^b + x* ^f^25. A ratio of greater inequality is increased, and a ratio of The proof is similar to tliat given in the last Art. Thus the dupliJ'il'^fi^f °:.! r™*^" ^--ff-";- ratio a^ : i^ is a . o , and the tnphcate fii^nW^"'" ^^ *^''' ^"^^'^'^"« «"^^ *h^t the ratio of the lust to the second equals the ratio of the second to the third then the ratio of the firsf tr^ fK^ +u- j • x, , •-" ^"® ""rd, the first to the second. '"' " the duplicate ratio of Let a, A, c be the three quantities, the" ^ = -,and ... (lY^- *,« which proves the proposition. i' t.??" ^^ *^r ^ ^''"' quantities such that the ratio of the fir^t cate ratio of the first to the second ^' *"P^'- The proof is similar to that given in the last Art. 30. Th-^ c,.uj.._.._ , . _ ...«.-u«p„catc and Subtriplicate Ratios of two RATIO. ii / quantities are the square and cube roots respectively of their ratio; but these terms are now seldom used. 31. The Inverse Ratio of two quantities is the ratio of the second to the first. It may easily be shown to be the same as the ratio of their reciprocals; lience inverse ratio is oiian called Reciprocal Ratio. A- 32. A ratio is increased by compounding it with a ratio of 'greater inequality^ and diminished hy compounding it with a ratio of less inequality. Let a:b he compounded with x : y, then the resulting ratio ax : iy > or < a : 6 according as a; > or < y. -n, ax a a(x — y) by b~ by ' and this result is positive or negati^ o according as a? > or •< y, which proves the proposition. FUNCTIONAL NOTATION. 33. The symbol /(x) has already been used to denote a func- tion of X. In the same way /(x, y) may be used to denote a function of x and y, f(x, y, z) to denote a function of the three quantities a-, y and z, etc. The form of a function is the par- ticular manner in which the quantities are involved. Different functions of the same quantities are denoted by using different letters before the brackets enclosing the quantities; thus F{Xy y) and f{x, y) denote different functions of the same quantities, x and y. Sometimes a subscript or other distinguishing mark is used, thus, F^{x, y), F^(x, y\ etc. Again, if the form of F{x^ y) be given, F{m, n) may be written by changing x and y in the given expression into m and n respec- tively; thus, i£ F{x^ y) denotes a3^+2hxy + bf, then F{m, n) de- notes am^ + 2hmn + bn\ -^ 34. If F(x, y) denotes a homogeneous function of x and y of r ' dimensions^ and if for x and y in this function we substitute mz and nz, the result will be a^i^w, n). i 22 HIGHER ALGEBRA. = «m'-«'- + Am'-'w;^'- + cm- V^r ^ etc = «'-(«W + bni'-\+ cmr-2^^ + etc ) ' function, then for x, y , 1 tit *.' -• "'" '"'3 *«™ » this *(«)"M>> twr J u°" '" J^'-«=«»"«.» with then exercise him/elf'bvw -7 ."*'"''"'"''"''"^- Heshould Hnd inCnaed in t^ f^i^^^^t" ^rr''"'"'"' °' ''''' by independent work. ^ ^*'" '" e^h case :V36. If there be two eqtud ratioi ih. ™/. ^ '^oftHe ZZ :Z '° ''' "'"" "-^ '^ -->-^^ 0/ the Let I _ ^ be the equal ratios; /-(a, «), y(„, «) ^h, homogeneous :rr:r °' ^ ""-^-^ "- --> ™«o =». tHe„ „=:: Then ( Art. 34, Cor. 2. and Therefore ^^(^^_J^(c, d) BATIO. 23 f 37. If there be any number of equal ratios, the ratio of any homogeneous futiction of r dimensions of the antecedents to the same function of the consequents is equal to the r*^ pcnver of one of tlie equal ratios. ace > Let - = -=-, = ..,. be the equal ratios. Let F{a, c, e, ....) be any homogeneous function of r dimen- sions of the antecedents; F{b, d,f....) the same function of the consequents; put each of the equal ratios =m, then a = mb, c = md, etc. Then ^(«, c, e,....) ^ m\F(b,d,f,....) ^ /Art. 34, F{b,d,f....) 'F{b,rf,f„„) -^' Icor.l. which proves the proposition. EXERCISE II. 1. Write down the duplicate ratio of 5 : 7 and the subdupli- cate ratio of 289 : 400. 2. Which is the greater of the ratios, 9:10orl0:111a;:ar+l or x+l.x + 2t a^ + b^:a^ + b^ or a^ + b^:a-]- hi 3. Compound the ratios 8 : 11 and 33 : 40; a:b, 6 : c and c : a. 4. Two numbers are in the ratio 5 : 7, and if 33 be added to each the resulting numbers are in the ratio 8:11. Find the numbers. 5. Find the ratio of .r to y from each of the following equations : (1) ax-by = cx + dy. (2) Zx'^-Txy = Qy\ (3) 2^2 _ 5^y ^ 2f = 0. (4) mx + ny = a(mx - ny). 6. The sum of two numbers is 100, and their ratio 7:13. Find the numbers. 7. If 5 men and 6 boys do as much work as ? men and 3 Ko"3 and 40 men and 15 boys together earn |114 per day, find the wages of a man per day. ^ ' f I 24 K HIGHER ALGEBRA. -Wi„ eaoH the. J^ it^^W ^Torr sT ^r many chapters are there in each ? "^^ ot JU 87. How H eUtr :1L^ :f ^r *"> -^ *- <" »-' -^o » : . .. „a.e oa^Uw ::f " ^ ' -"^ * ^ '■ -" «<"-'. "-» <. : c i. the dupH- 14. Ua:b==b:c, then l'^- Which IS the greater ratio, « and i having like signs ? + ^ • « + a 6 + 6*, I 16. If «:i + c = w:riandi-c4.«-«. « j ., ,^ir. The .ates of two trainj aIhTb ^^"""^ ^"*- ength. „( their .journeys are IpT li ZT' Vt *'■"' longer to make its journev th„„ ■. T ^""^ ^ * h""™ of each. '^ J""™/ tl««« It does train A. Knd the time -- tfm:Copp:z !:r zr '^-^ '"-^ ^^^^ - ^-o ratio of the timeXylale !ft '""' ""^' ''"'^ *''»' *he is n' : ml ^ '" *'*<"■ "^^ "°fe- to finish the journey p-al!d^::erp:l•rtl:f^!r'""^^"^''»«*''-p^-> tively. Find irj; ~'°° 1"'''°««» "^ ""J 6* lbs. ■■esnec balanoe. " '"*" ""'' '"« ™tio of the arms of the RATIO. 25 (V20. Each of two vessels contains a mixture of wine and water. A mixture formed of two measures from the first and one from the second contains wine and water in the ratio 56 : 79 but if one measure be taken from the first and two from the second the ratio is 58 : 77. Find the ratio of wine to water in each vessel. 21. I£7n gold coins are equal in weight to n silver coins, and p of the former equal in value q of the latter, compare the values of equal weights of gold and silver. 22. Urn gold coins placed side by side reach as far as n silver ones, and p of the former are together as thick as q of the latter, and the values of equal bulks of gold and silver are as r:*, com- pare the-v^alues of a gold and a silver coin. 23. A street railway runs along an incline, and the ratio of the rates of a car up and down is 2 .■ 3. The cars leave each terminus every ten minutes. At what intervals of time will a car going up meet the successive cars coming down, and vice versa? 24. A straight line is divided into two p&.rts in the ratio p : q, and again in the ratio r : s. The distances between the points of section is a. Find the length of the line. (y 25. A straight line is divided into three parts in the ratio piq.r. Find the ratio of the segments into* which the middle point of the line divides the middle part. 38. A certain class of equations which occur very frequently in the higher mathematics may conveniently be discussed in con- nection with the subject of this chapter. Suppose we have given the huigle equation a.r + iy=0, the values of x and y are evidently mdeterminate, since any value whatever may be assigned to one letter, and then a corresponding value may be obtained for the other. If, however, the different solutions be examined it will be found that the ratio of the values of x and y is constant, what- ever value may be assigned to one of the letters. The equation may be written «(^") +6 = 0, from which ? = - ^. In fact, the original equation is not properly an equaUon between two un- V 26 HIGHER ALGEBRA. ^ knowns, x and y, but an equation with om unknown, m., the ratio X : y. This fact will be more clearly perceived if we take the two equatio^s, ax^hy = {i and a'x + h'y = % when it will be found that the second equation will not assist in determining exact values for x and y, but will be inconsistent with the former unless ^ = ^ . If this condition be fulfilled, the second equation is a mere repetition of the first ; if not, a: = and y = is the only solution. "; 39. Between three quantities, x, y and «, two independent ratica exist, viz, x:zB.udy:z. A third ratio, x : y, might be written, but Its value is dependent on the other two. Both ratios may conveniently be expressed thus, x:y:z, which form has the addi- tional advantage of representing the ratios of any two of the three quantities. An equation of the form ax + by + cz = may be considered an equation between two unknowns, viz, the ratios ar : « and y : «, as will immediately appear from dividing through by z. If, then, two such equations be given, th • values of the two ratios may bo determined. Ex. 1. — Given «a; + iy + os = 0, ) a'a; + hy + dz = 0, j *"* ^""^ *^® ^^*^°^ x-.y.z. Dividing each of the equations through by z, and solvin- for X y - and - , we get z z X and he' - h'c z ab'-a'b y ca' - c'a z ab'-a'b' These results may be written in the more symmetrical form, « y z hc'-b'c ca'-c'a ab'-a'b' H^-.toh gives the value of the ratios required, 3 H I RATIO. 27 Ex, 2.— To find the condition that the equations, ax-\- bi/+ csi = 0,' a'x + b'y + c'z = 0, may be satisfied by the same values of x, y and z. Writing the ratios x-.y.z from the second and third equations we get X y b'c''-b''c' c'a"-c''a'~7Un^" Now divide the terms of the first equation by these fractions in succession, which is merely dividing through by the same quan- tity, though in. different forms. The result is a{b'c'' - b"c') + bic'a" - c'a') + cia'b" - a'b') = 0. If this condition be fulfilled the equations are satisfied by x = k{bV-b''c'), y.= k{c'a''-c"a'), z = k{a'b'' -a'b'), where k is any multiplier, since it is evident that the ratios of these values are the same as the ratios x:y:z originally given. If the above condition be not fulfilled, then the only values which will satisfy the equations are .r = 2/ = « = 0, which evidently satisfy any similar set of equations. 40. The examples of the preceding Art. are of great impor- tance, and the student should be able to write the ratios from any similar set of equations without going through the succes- sive steps of tlie solution. The res-Its may easily be remembered as follows: Write the equations one above the other, then omitting the coefficients of each letter in turn the coefficients of the other two form a square as below : i'.c c, a, c, a', a, b, «', b\ the letters a, b, c, a\ b', c\ following each other in the usual cir- 28 HIGHER ALGEBRA. cular order. The letters of these squares form the denominators of . y and . respectively by taking the difference of the products of heir dmgonals, heg^nn^ng M the Hagonal drawn don,nwards to the r^ght The signs of the coefficients must be taken in con- ne'^tion with the coefficients themselves. Ex. i._Write the ratios x^y.z from the equations, Result: y (-3)(-l)-(2)(l) (l)(l)-(2)(3T) = (2)(2)rrp3^^ or 1 3 7 Ex. 2. — Solve equations: ar + y + «=:0, Writing the ratios from first and second equations, h-c c-a~a-b' Dividing the terms of the third equation by these fractions, a\h -c) + b\c - a) + c\a -b) + {a- b){b - c)(c - a) . ^11^= Q. Dividing through by - (a - b)(b - c)(c - a), ^--^ = or x = b-c, from which the values of y and .may be written, since the ratios x:y:z are known. Sometimes it is convenient to combine other quantities with X, y and z. and th^n f^ iir«,'f^ au„ -._j.-. n .. . • - - ; •'-'^- «^nc Litios or tne resulting expres- sions as m the following example ; a i' *■ lenoniinators the products I dowmuards aken in con- is. -3)(1)' •actions, X the ratios ;ities with ig exprea- HATIO. 29 jEx. 3. — Solve equations: x + y + z^a + b + c] ax + by + cz = ab + be + ca, (b - c)x + (c - a)y + (« _ b)z = 0.^ The equations may be written, (ar-i)+ {y-c)+ {z-a) = 0, . a(x-b) + b(y-c) + c(z-a) = 0, (^-c)(x-b) + (c-a)(y-c) + (a-b)(z~a)^bc + ca + ab^a^-b^-c'. From first and second equations, x~b y~G z—a c -b a -c b — a* and then from third equation, b~c x-b b+e Therefore X —b ' 2=- or x = - 'i ' from which the values of y and z may be written from symmetry. 41. Commensurable Quantities are those which have a common measure, or those which are capable of being expressed Ire tIT 1,*'rr"' ""*• I^co^^niensurable Quantities are those which have no common measure, or are not capable of being expressed in terms of the same unit. A good example of incommensurable quantities is furnished 1 luM ""^ "" '•^"^'^ ^"^ ^*" diagonal. There is no unit of ength that is contained an exact number of times in each. If the side be divided into 10 equal parts, the diagonal will contain more than U such mrt,s. l.nf lo^c +!.„>, ik. :e .> v. ■, ^ ■,/^^ , '■ ' — '^""" ^'^) " it oo aiviaea into 100 equal parts, the diagonal will contain more than 141 but less tnan 142 such parts, and so on to any extent. Similarly, if the ■f j so BWHEB ALOBSaA. ■ diagonal be divided into an equal number of exact parts, the side wil. not contain an exact number of such parts. All this is briefly expressed by saying that the quantities are incommensurable. 42. The relative greatness of two magnitudes is in no way dependent on the manner in which they may be represented by symbols. The diagonal of a square axlmits of being compared in regard to its length, with the side, even though they can not' be represented numerically in terms of the same unit. The defini tion of ratio, therefore, of Ait. 21 is not strictly appropriate for incommensurable quantities, and it is in this particular that it is inferior to that of Euclid. But though the ratio of two incom- mensurable quantities can not be exactly expressed by numbers. It can be expressed to any required degree of accuracy, as is shown in the following Art. The ratio between two incommen- surable quantities is called an Incommensurable Ratio. 43. If two quantities are incommenmrahk, afracti(m may he found which will represent th^ir ratio to any required degree of accuracy, ^ •' ^ For let a and h represent the two quantities; let b be divided into n equal parts, and let x represent one of those parts; then b==nx Also let a be greater than m.r, but less than (7^+ 1)^- then - > -, but < -_; then the difference between "^ and - J b n is less than -. Therefore by taking n large enough the fraction m - differs from the exact ratio of a to i by less than any assign- able quantity. In such examples ^ and "^ are said to be the Hmits be- tween which the true value of the ratio lies. 44. Two im:omr>unsurahle ratios are equal providing tUy always he between the same limits /m?,,^.,.^ ^, 7/ .1, ,.^ I . ^, ... ' —''■^■^■-' cnttAiL crie ainere'nee between those limits may he RATIO. 81 in no way resented by ompared, in can not be The defini- ropriate for ar that it is two incom- •y numbers, iracy, as is incommen- latio. iofi may he d degree of be divided 'arts; then I {m-\-\)x', « , m 1 r and — o n le fraction ny assign- limits be- Ung iliey differeyice For let a : 6 ajid c : c? be the two ratios whose values each lie m wn- 1 between — and ; then the difference between those ratios n n 1 is less than -, and by taking n large enough this difference may be made less than any assigned difference between the ratios; and since there can be no assigned difference between the ratios, they must be equal. EXERCISE III. ^M. Given ar + y + « = 0,\ 2^ + 3y + 4« = 0j ^"^ *^^ ^^^^'^ ^= 2/:«. *^. Given a? = ay + is;,) ^ , t fi^d the ratios xxyxz. y = bz+c.r,) ^ *^. Solve equations: 2x + y-z = 0y ar-22/-3« = 0, x^ + xy + y^ = 8i. ■\^ 4. Solve equations: i^ ^t% 2x-3y + iz = 0, (> Zx-y-2z = 0,"^^ ar' + f + ^^.biSd. .1^ fV .X e^. If X — a y — h z — c then each fraction = 6. If and then I m n and (x+a)l+(y\-h)m + {z+ c)n = p P z X aif,:^ "^ h{c-a) "^ c{a - h) " ^' aj^-cf _ h{c - af c{a- hf b-c c—a a—b y . « X y ^ ds fllOHER AL6SBRA. 7 If ^-yi__y~zx x{\ - yz) yfnr^* ^1 y b© unequal, then each fraction = 8. If then _^-^y . Ill and {h ^ c)(^ > y.) + (, _ aW - zx) + (a - b)(^ - :ry) = 0. a? C_ 9. If and then ~ and V z *+c c+a a+6 X y z X J ~ p — - o-c c-a a-b 1 y C 10. Solve the following equations: f c 11. r <^^ + by + cz=(c-b)x + (a-c}y + (b-a)z ^a' + V' + c'-ab-bc-ca. x-¥y + z = Z{a + b + c), +{c-a)y + {a-.b)z^bc + ca + ab-a^^K^-^, ^^ + cy + az = cx + ay + bz = a^ + b^ + c\ x + y + z = a + b + c. ^^' " .« + y + z = a + b + c, ax + by + cz==ab + bc + ca. (f' + c')x+(c + a)y + (a + b)z = a'+b^ ^ + c^ + ab + bc + ca. ftATlO. 3S 14. ax + by by ■{• cz cz + ax a-b b-c c-a * €fix + Iy^i/ + c^z + (a - b){b - c)(c - a) = 0. ir 2a? + 3y -4z _ 33r + 4y - 2g 4x + 2i/-3z x + y-z x + 5 5x 4x~-n 6 ' i> 16. a'x 2/V />3. '2/ z^x'^ c^z a?2y2 1 17. If Val±Vf>m±:Vcn = 0, the two values oi y:z obtained from ^^-+- + -=0, X y z 2'^ lr + my + nz = Oj will be equal 18. Find the condition that the equations, ax + ?iy + gz = 0, hx + by+fz = 0, gx+/y + cz = 0, may be satisfied by the same values of ^, y and z, 19. Find the condition that the equations, ax + cy + bz = 0, bx + ay + cz = 0, cx + by + az = Of may be satisfied by the same values of x, y and z. 20. If the equations, ax + by + cz = 0, a'x + b'y + c'z = 0, a''x + b''y + c''z = Of are satisfied by the same values of x, y and z, tlien ax + a'y + a''z = 0, bx + b'y + b''z = 0, cx + c'y + c"z = 0, are also satisfied by the same values of x, y and z. x » y 21. If = a. = A, = c. y + z ' z + x ' x + y find the relation between a, b and c, and show that x' r ^ I a (1 - be) 6(1 - ca) c(l - ab) ^^ d4 . 22. If X: prove 23. If UlOHER AIQEBKA. •cy + i«, y-a« + ca., z=^bx ar> + «yi y' l-a» l-As'ni l-cr*' 26. If d; y (y+«) h{z^.x) ^^T^' a-b b~c c~ c~a «y/ 2/z Z2/a_7,\ . y^/; , zx. X y then and 26. If ^ then and ^ - n 27. If «X^..r+cZ=0 and «.r+M'-...^=0 where then ^2+ ^2 + ^2= K(Ac^_-V) + ^>2K ^/i/?. _ A ^^2 ,~7TI tt; ^.-4 — LjLL CHAPTER III. PROPORTION. ,45. Proportion is the equality of two ratios. Four quanti- 'iies are said to be in j roportion when the ratio of tue first to the second equals the ratio of tht; third to the fourth, and the quan- tities themselves are called proportionals. The first and fourth quantities are called Hxtrctnes, and the remaining two are called Means. 46. The equality of two ratios may be indicated in various ways. Thus, ii a:b and c : d are the two ratios, a:b::c:d (read, as a is to i so is c to c?), a : A = c : <7, or r = ti indicates that the ratios are equal and that the four quantities are in pro- portion. Similarly the equality of three or more ratios may be expressed, thus, a : i : : c : , =-, or <, nD, mA >, =, or <, nBf I for all positive integral values of m and n, then A, £, C, D are said to be proportionals. OTT we shalTiwAv show" that quantities which are proportional according to the algebraic test are proportional according to Euclid's test, and conversely. -# 1. Let a, i, c, d be proportionals algebraically. Then \'^\ by definition. Multiplying each fraction by — n we get ma mc nb nd' f Now, ma and mc are any equimultiples of the first and third, and nb and nd are any equimultiples of the second and fourth, and from the principles of fractions wc >, =, or <, nd, according as ma >, =, or <, w6, which proves the proposition. Let a, b, c, d be proportional accordi sometrical PROPORTION. 41 definition, then shall they be proportional according to the alge- braical definition. a a For if r be not equal to -, let - be the greater; and let — be n m less than r, but greater than -. a Then, since and since a n T > — , .*. ma > no; m ' en J < — , :. Tnc < nd, a m Now, of the four quantities, a, b, c, d, of the first and third equimultiples ma and mc have been taken, and of the second and fourth equimultiples nb and nd have been taken; and the mul- tiple of the first is greater than the multiple of the second, but the multiple of the third is less than that of the fourth, which is contrary ic the supposition that a, b, c, d are proportional accord- ing to the geometrical definition; therefore - is not unequal to c b ^, «,e., they are equal, which proves the proposition. EXERCISE IV. 1. Find a fourth proportional to 3, 5 and 15. , 2. The second, third and fourth terms of a proportion are 12, 41 and 61 J; find the first. 3. Find a third proportional to 1 + V' 2^ and 3 -H 2 V\ and a mean proportional between V^ 7 - V 5 and 1 1 \/ 7 -J- 13 Vb. 4." What number must be added to each of the numbers 1, 3, 5 and 8 so that the results will be proportional? 5. Find a number which, added to 1 and to 11, will give re- sults between which 12 is a mean proportional. 6. Given that x ■{- y: x — y.: a ■{■ b \ a - b^ and wi i portional between x and y, find x and y. mean i 42 HIGHER ALGEBRA. B 'I ■I y^" \ c 7. Three numbers are in continued proportion; the sun of the greatest and the least is 51, and the sum of the two greatest is 60. Find the numbers. C^ 8. Given that the work done by a;- 1 men in ar+ 1 days is to the work done by ar + 3 men in a: - 2 days as 20 : 21, find x, 9. If four quantities are in continued proportion, the difference between the first and last is more than three times the difference between the other two. ^ 10. If (a'^+b^)(b^ + c') = (ab + bc)\ then «, b, c are in continued proportion. 11. If a, i, c are in continued proportion, *hen a + mb:a-mb::b + mc:b-mc, and (c + ^) • (^ + I J '" ^ '^*^'* °^ equality. 12. What must be subtracted from each term of the ratio a : b that the resulting ratio may be the duplicate of the original ratio 1 >U3. Find two numbers whose sum, difference and product are proportional to s, d and p. b' c' d* ~ ^^ proportionals, ' then b* + a^c" : b* - aV y.d* + c'e^ : d* - cV ; and if a, b, d are also in continued proportion, then <^ = e. (,^15. If 6 + c + c?, c + c?+ a, 0? + a + i, a + 6 + c are proportionals, then i' + 6c + c2 = a2 + a(f+cf2^ ^ 16. What must be added to each of the four quantities a, 6, c, d so that the results will be in proportion ? Examine the case in whicha + o?=i + c. ^ 17. Ifa + i:7n. + 7t::m-w:a-6, *^hen a + ni:b + n::b-n:a-m; m(\ if a is greater than i, then b is greater than n. PROPORTION. 43 ^18. If a, 6, c, d are proportionals, (a - b)(a - c) then a+d=b+c+ a J 19. If a, b, c, d are proportionals, and if a is the greatest, then d is the least; also, a + d>b + c and a^ + rf?> i^ + c^. O.20. If the ratio of the difference of the antecedents of two ratios to the difference of the consequents is measured by the sum of the measures of the separate ratios, the antecedents are / in the duplicate ratio of the conseqijents. . . - ' *^'m ^ Jc/U, -J r,>-i 'i A 21. If ar and y be such th^*^^en added respectively to the .awtecedent and consequent/^ the ratio a : h, the resulting ratio is >-ryT^,t^^ reciprocal of that formed by adding them to the conwequent y ^ and antecedent, then either a = 6orar + ?/ + a + 6 = ^ 22. lii{a-k'b-\rc + d){a~b-c + d) = {a-b + c-d){a + b-c-d), then a: b::c:d. ^ 23. If {pa + qb + rc + sd){pa.r-qb-rc + sd) — {pa-qb + rc-sd){pa + qb—rc-sd)y then bc:ad::ps: qr, w^\ ^ and br-.pdi'.as'.qc; Q0 and if either of the two sets, a, b, c, c?, or p^ q^ r, «, are propor- tionals, the others are proportionals also. ^. 24. Sold goods for $24, losing as much per cent, as the goods cost; find the cost. What should be the cost for the selling price to be as great as possible ? 25. The time >^hich an express train takes to travel 180 miles is to that taken by an ordinary train as 9:14. The ordinary train loses as much time from stoppages as it would take to travel 30 miles without stopping. The express train loses only half as much time as the other by stopping, and travels 15 miles an hour faster. What are their rates respectively 1 5—26. To 300 lbs. of a mixture containing 2 parts of zinc and 3 of copper and 4 of tin was added 200 lbs. of another mixture of p)0 1 I ■d 44 HIGHER ALGEBRA. the same metals, when it was found that the proportions were now as 3, 4, 5. What were the proportions in the added mixture ? 27. Each of two vessels contains 10 gals, of alcohol and water, the first in the ratio of 3 : 2 and the second in the ratio 1:4; how many gallons must be poured from the first into the second, and then the same amount from the second into the first, to leave 5 gallons of alcohol in the first vessel ? 28. The first of two vessels is filled with wine and water in the ratio m.n; the second in the ratio p : q Their contents being mixed, the resulting liquid contains wine and water in the ratio r:8; find the ratio of the volumes of the vessels. 29. If a, b, c are in continued proportion, then {a-hy:h{a-c)::a{b-c):c{a + h)', a\a-h + c):a? + ab + b'^'ui?-ah-k-h'^'.a-\-b + c', ^2 4i {b+cf {c + af (a + by 0- -, c — a a — b a —c (a + b + c). Q^ 30. If a, b, c, d are in continued proportion, then (a-c)(b-d)-(a-d)(b-c)=={b-cy; (b-cy+(c-ay + {d-by=={a-dy; (ad + bc)(a + b + c + d){a -b-c + d) = 2(ab - cd)(ac - bd). 31. Brass is an alloy of copper and zinc; bronze is an alloy containing 80 per cent, of copper, 4 of zinc and 16 of tin. A fused mass ot brass and bronze is found to contain 74 per cent, of copper, 16 of zinc and 10 of tin; find the percentages of copper and zinc in the composition of brass. 32. A and B are partners in a business in which their interests are in the ratio a : b. They admit C to the partnership, without altering the whole amount of capital, in such a way that the interests of the three partners are then equal. C pays $c for the privilege. How is this sum to be divided between A and B, and what capital had each in the business originally 1 CHAPTER IV. TARIATI®N. 58. The object of the present chapter is to present the principle of proportion in a slightly different form, and one which is spe- cially adapted to complicated and intricate problems. The diffi- culty usually experienced by students in this method of treatment of the subject will be removed by giving careful attention to the meaning of the technical terms employed, as explained by a few simple examples. 59. The concrete magnitudes to which mathematics are applied are so related to each other that a change the one frequently produces a corresponding change in another, or, in mathematical language, one is a function of the other. Thus the circumfer- ence, the surface and the volume of a sphere will each be changed if the length of the radius be changed; i.e.., they are each func- tions of the radius. The ratio of the circumference to the diam- eter, however, will not be changed by changing the radius; the value of this ratio is therefore a "constant," whilst the radius, circumference, surface and volume are "variables." If we con- ceive of different values being given in succession to the radius, it may be called the "independent" variable, whilst the other quantities, from their values being dependent upon the value of the radius, are called " dependent " variables. 6§. The value of one quantity may depend upon the value of several others; thus the area of a triangle depends upon, or is a function of, the base and altitude. When one quantity increases another may diminish; thus the time necessary to travel a given y 46 HIGHER ALGEBRA. l\ in ^^^yNflJ If distance diminishes as the speed increases. The number of ways in which the value of one quantity may be connected with, or be dependent on, others is without limit. In any particular problem the exact nature or the dependence or connection must first be clearly conceived in the mind, and then accurately expressed in mathematical symbols. 61. If a single equation be given, containing two unknowns, X and y, we may give any value we please to one of them, and then corresponding values of the other may be determined. In such cases either quantity may be considered a function of the other, and the quantities themselves are variables. Consider the following simple examples : * y = 2x, y = ix\ y = ax + h, y = ax'^ + hx + c, y = 2'. In each case the value of y is known when that of x is known; we therefore say that x and y are variables, that y is a function of X, and that x is the independent variable, and y the dependent variable. 62. We have now given a number of examples of both con- crete and symbolical quantities, of which the value of one varies {i.e., changes) when the other varies or changes, and therefore, in popular language, one may be said to vary as the other. But the word "variation," in mathematical language, is restricted to one particular kind of change in value, or functional dependence be- tween two quantities, which we shall now proceed to explain. 63. Variation is an abridged method of indicating proportion; its precise meaning is determined by the definition of the follow- ing Art. ^ r "*• ^^® qp^«?f y is said^p vary dirpctly as another when the >i«,tio, the other is inext increase the height i)-4 to DF. Denote this height by z, and the area of the triangle FBE by y' ; then -^ = -,. From these ^. , . 1/ xz y ^ i- / ,— > *vnicn anoirVa tnao tne us-ca varies as y xz the product of the height and the base. 50 HIGHER ALGEBRA. r % .^Z'J^ .v'^natxons ™ay be inverse, or one may be direct rushed by the change of pressure of a gas when the volume and sure, p of a ga^ vanes as the '-absolute ten.perature," t when the volume, ., ,s constant, and inversely as the volume when the temperature is constant; that is, /> oc < when V is constant, *^^ ^ =^ - when i! is constant. From these equations;, c ^ .hen . and . are both variable, and }y actual experiment this is found to be the case. 72. I/y ex: cc, then amj hanioffeneous function of x and y varies 22^^y otker kom.,ene.us function of tke sa.ne rirr.,er ofIZZ Let F{x, y\ fl^c, y) denote any two homogeneous fu. ions of ^andy, each of .-dimensions; and. since^::.^, • y^,^^ ""^ Then r^(fLy)^jXgLi!^),^'-.i^(l,m) F{\, m) /(^. y) f{x, mx) x\ f{l,m)~ f{\,m) ^ ^ <^o^tant. Therefore ^(^, y) oc./(:r, y). \y^^' ^\ ^"^ ^'1""'*^°" °^ variation exists between two homo- ^hr;L""^ of the same number of dimensions Z and ^ Let F{x, y), y(^, y) denote two homogeneous functions each of r dimensions, of which one varies as the oMier Let y = ,n.r, then we have to show that the value of m does not change when x and y change. «|n- ^(-,y) «:/(., y), :. Fi.,y)^k.Myy Therefore ^(r, m.r) = >{:./(.r, «^.r); .". ^^^ym) = k.f{\,m). VARIATION. 51 Hence the value of m is independent of the values of x and y, and is therefore a constant; /. y f--hen 2. Given that z varies jointly as .. and y, and when x = 1 v/ - 6 and ^=16, find the value of y when ^=150 and :r = 6. 3. Given that 4x + 6y oc 2.. - 6y, and when ^ == 10, y = 1 fi„d the ratio .tr : y. ^» y i> una ^4^ Given that j, oc^. + ,, and whea .= ,, 2, ^=5, 7. find the 5. Given that , varies directly as :. when y is constant and mversely as y when . is constant, and when .= 16, yTu ^ 2 =40, find the value of s when ;.«_ 5x3, -6y'. 6 Given that y varies as the sum of two quantities, one of wh.ch .s constant andthe other varies inve Jy as ., and li - X, , , X, _ i^, o^ una tiio value of x when y = 10 1 .;h ^f VARIATION. 55 C 7. Given that ip cxz a'' - x\ and that when ;r = 0, y = ±A, find the value of y when x = Va? - h\ ' ; ^ 8. Given that « oc <» when / is constant and s oc/ when t is constant, and when t=l, 28=/, find the equation between s, f and t. m (p. The surface of a sphere varies as the square of the radius, and its volume varies as the cube of the radius. Find the radius of a spliere whose volume equals the sum of the volumes of spheres whose radii are 3, 4 and 5 feet respectively, and compare its sur- face with the sum of the surfaces of the three spheres. 10. If s" oc ve and t?* oc sf" when / is constant, and «« oc t^y and ^ oc s/* when t is constant, show that v" ocfs when all vary. (Jl. Given that y varies as the sum of three quantities, the first of which is constant, the second varies as x, and the third varies as x\ and that when x = u, 2a and 3a, y = 0, a and 4a, find y when x = {n+\)a. /J 2. Given that z varies directly as x and x varies inversely as y, and that when ar = 4, y + ;s = 340, and when x=\, y-z=1275, for what value ~>i x ia y = z'i C 13. If y oc a:, then x~y oc x + y and r^ + y' ^ xy(ax + by). el4. If yoc2;ocrr, then x^ + y^ + ^ oc xyz oc (x + y + zf. ^ 15. If ax + byaccx + dy, then y oc a: and x^ + y^ocxy. ae. If x + yocz and z + xocy, then xocyocz and ^IZ + yz + zxcxzx^ + y^ + z^ CJ.7. If y'^oczx and z"^ ex xy, then x^ - yz c< (x'^z)l aa^. If xocy% focz\ ^cxzu' and u' oc v\ then xyzu ocv\ 19. If xocy + z, xzocu' + y'i and y'cxizix + u), then (^ + y){y + z){z + u){u ■\-x)oc xyzu oc aa:* + by' + (c;^" ^ ^^^ ^ ^^2^,^ 20. If X, y and ^ be variable quantities such that y + z-xis, constant, and that {x + y - z){z + x ~y) oc yz, then x + y + zocyz. 56 HIGHER ALGEBRA. 21. If ix + y-\.z){x-\.y -z){x~y + z){- X + y+ z) oc, x^yi^ then either ar^ + y^ocz^ or x"" + y"^ - z^ oc xy. Give a geometrical in- terpretation to this example, v <^i/22. A locomotive engine without a train can go 24 miles an hour, and its speed is diminished by a quantity which varies as the square root of the number of cars attached. With four cars its speed is 20 miles an hour. Find the greatest number of cars which the engine can move. 623. If the attendance at church varies directly as the preacher's ability, and inversely as the square root of the length of his ser- mon; and if 240 and 350 persons attend A's and B's churches when the sermons are 49 and 36 minutes long respectively, com- pare A's and B's ability. 24. The time of the vibration of a pendulum varies directly as the square root of its length, and inversely as the square root of the force of gravity; and gravity varies inversely as the square of the distance from the earth's centre. Find the height of a tower on whose top a seconds pendulum loses one second in 24 hours. If the length of the pendulum be 39.139 inches, how much must it be shortened to make it keep correct time in the elevated position ? >l25. The value of a diamond varies as the square of its weight. Three rings of equal weight, each composed of a diamond set in gold, have values of o, b and c dollars, the diamonds in them weighing 3, 4 and 5 carats respectively. Find the value of a diamond of one carat, the value of the workmanship bf ii g the same for each ring. 26. The value of diamonds varies as the square of their weight, and the square of the value of rubies varies as the cube of their weight. A diamond of a carats is M'orth m times the value of a ruby of h carats, and both together are worth %c. Find the values of a diamond and of a ruby, each weighing n carats. -^27. The velocity of a railway train varies directly as the square VARIATION. 57 ^y', then trical in- miles an varies as four cars 3r of cars reacher's f his ser- churches elv, com- rectly as e root of e square ght of a id in 24 lies, how e in the 1 weight, d set in in them lue of a ?h g the weight, of their lue of a e values root of the quantity of coal used per mile, and inversely as the number of carriages in the train. In a journey of 25 miles in half an hour with 18 carriages 10 cwt. of coal is required- how much coal will be consumed in a journey of 21 miles in 28 minutes with 16 carriages? 28. The consumption of coal by a locomotive varies as the square of the velocity. When the speed is 1 6 miles an hour the consumption of coal per hour is 2 tons; if the price of coal be $5 per ton, and the other expenses of the engine be $5.62^ per hour, find the least cost of a journey of 100 miles. O^. The square of the time of a planet's revolution varies as the cube of Its distance from the sun; find the time of a revolu- tion of Venus, assuming that the distances of the Earth and Venus from the sun to be 91^ millions and 66 milHons of miles respectively, and taking our year to be 365 days. ^ 30. The attraction of a planet on its satellite varies directly es the mass (M) of the planet, and inversely as the square of the distance (B); also, the square of the time (T) of a satellite's revo- lution vanes directly as the distance, and inversely as the force . ^!r'^^T' ^^ '''" '^^' ^' ^""^ '^^' ^^' ^^ ^^« simultaneous values of Jf, i>, T respectively, prove that dl df *2''2 "-2 Hence find the time of revolution of that moon of Jupiter whose distance is to the distance of our moon as 35 : 31, having given that the mass of Jupiter is 313 times that of the Earth, and that the moon's period is 27.32 days. square CHAPTER V. ARITHMETICAL PROGRESSION. 78. A Series is a succession of numbers or quantities which are formed in order according to some definite law. Thus 1, 2, 3, 4 is a series, the law of formation being that each number is obtained from the preceding by adding a unit. Also, a + h, a^ + b\ a^ + h^,, ,, is a series of which the law of formation is evident. 79. An Arithmetical Series, or an Arithmetical Pro- g^ression, is a succession of numbers which constantly increase or constantly decrease by a common diflference. Thus each of the following series is an arithmetical progres- sion : 1, 2, 3, 4, 5 ... . 20, 17, 14, 11 .... a, a + d, a + 2d, a + Sd a, a- dy a -2d, a — 3d The words " arithmetical progression " are briefly denoted by the letters A. P. 80. Each of the successive numbers in a series is called a Term ; the first and the last terms are sometimes called Ex- tremes, and the intermediate ones. Means. In an arith- metical series the difference between the successive terms is called the Common Difference. 81. Any arithmetical series may be represented by a, a + d, a + 2d, a + 3d . .., i. ¥ ARITHMETICAL PEOGRESSION. 59 in which a stands for the first term and d the common difference The series will be an increasinff or a decreasing one according as d is positive or iiegative. ' 82. In any arithmetical series there are five quantities to be considered : The first term . . . . The last term . . . . The common difference . The number of terms . . The sum of all the terms which is denoted by a. " d n. l»t 2''d ard «, « + rf, u + 2d 'it^83. To find any required term in an arithmetical progression the first term and the common difference being given. Forming the terms in succession we have • a + M a + {n-\)d, from which we ol)serve that— Any term is found by multiplyvng the common difference by one less than the number of tli^ tenn, and adding tU product to the first term. This result is briefly expressed in symbols thus : w*Herm = a + (n-iy, °'' ^ = a + (n-l)rf. ^1^ ^V^^' To find the sum of any required number of terms of an A. I ., the first term and the last term being given. Write the series first in the natural order, then in the reverse order, and add. Then 'S'=a + (« + o?) + (rt + 2rf)+ .... (;„^)4-7 ^nd S=lHl-^-^{l-2d)^....\a^d)Va, I 60 HIGHER ALGEBRA. 2S=={a + l) + {a + l) + {a + l)+ .. == (a + 1) repeated n times . S" -(a + 0> the sum required. (« ■¥l) + {a + l) (2) 85. If any two terms of an A. P. bo given, the series is com- pletely determined. For, suppose the m»»' and w»^ terms are ^> and ({, Prom these two equations a and d may be found, and then the series is known. 86. To find the arithrhetical mean bettoeen two given extremes. Let a and h denote the given extremes and x the required mean, so that a, x, h are in A. P. x-a — h-x. -/. ^ ^^^ Then from which x = a + b y(-^' From the above it is seen that the aritlmietical mean of two quantities is half their sum; it corresponds to what is meant in common language by the word "average." ^ 87. To insert a given number of arithmetical means between ^iwo given extremes. Let a and h denote the given extremes, m the number of means, and d the common difference of the resulting series. Then, the total number of terms being m + 2, from the equation l = a-\-{n-\)d we get b = a + {m + \)d, b — a from which rf= m+1 (2) ARITHMETICAL PROGRftssiON. Therefore tlie required means are b 61 m+1 ' or w + 1 m + 1 ' m + 1 • ' 88. The equations, / = a + (w-l)(/, 0) (2) th m le e rr '""'^^" '' arithmetical progression. From them we can find any two of the five quantities involved when the other three are given, and are thus enabled to solv all J^l ble problems in the subject. ^^* By substituting in (2) the value of / from (1) we get ot solving problems in arithmetical progression: f"' '-^^"^^ « = 17, cf = - 3 and ^= 55, find n and /. Substituting the given values in the equation, ^=^{2a + (n-l)c/}, we get Simplifying, from which n 55=-{34-3(n-l)}. ^^^2 on.. , 1 -iz-w rt W = 5 or 7|. il i dt HIOHEII ALGEBRA. The value 7 J is not admissible, since, from the nature of the problem, n must be a positive integer. Then l=>a + (n-l)d -:17 + (6-l)(-3) -5. Bx. ^.— The sum of the second and fifth terms of an A. P. is 47, and the sum of the first four terms is 58; find the eleventh term. With the usual notation we have, second term => a + ) = (« + & + cXab + bc + ca) - Zabc = U{b{a + c) + ac}-3abc = 04", -• -■--.---. I'.n- ^r. vr^^Ociviuii, ARITHMETICAL PROORESSION. 68 BXBROISB VT. 1. In a series whose first term is 5 and common difference 2, find the tenth, fifteenth and one Imndrodth terms. 2. In a series whose first term is 35 and common difference - 3, find the eighth, fifteenth and /***» terms. 3. In the series — . (1) 4, 7, 10 .... , find the n*" term. (2) 8, 5, 2 .... , find tho n*"* term. (3) 2§, 2^, 2| , find the ninth and seventeenth terms. > (4) - 23, - 18, - 13 find the thirteenth and (w- 1)"" terms. M Intheserie8,17, 14, H ...., which term is -82, - 118,-419] b. The first term is 17 and the twentieth term is 150; find the common difference and the fortieth term. v6. The third term is 75 and the eleventh term is 131; find the twentieth term. i/7. The first term is 2a - Zh and the second term is 3a - 2i- find the w* term and the (2n - l)'" terra. Which term is (3/) + 5)a + 3joi? 8. Sum the series — u(l) 1, 2, 3, 4.... to 100 terms. (2) 8, 3, -2, -7.... to 20 terms. 1 3 ^ (3) 2 ' - 4 ' - 2 ... . to 24 terms. c(4) 2n-l, 2n-3, 2n-5 to n terms. /c\ 6 —12 ^^ ~7^' ^^^' --:-.... toSOterms. V 3 V3 .(6) w - 1, M- 2, n-i .... to 2n terms. •''•^^^^^^•S'^y^^Ms*""^^^^^ 64. mourn ALGfiBRA. ^ 9. Find the arithmetical mean between 33 and 17, 37 and - 5, m + n and m-n. •^10. Insert five arithmetical means between 2 and 20. ^11. Insert x arithmetical means between 1 and x\ (5 2. If m-n-l arithmetical means be inserted between, w' and w', what is the common difference of the resulting series? ^^13. The sum of the second and fourth terms of an A. P. is 30, and the sum of the third and fifth is' 38; find the first term and the common difference. tU. The eighth term of an A. P. is greater than the fifth by 24, and the sum of the sixth and tenth terms is 100; find the common difference and the w*** term. as. The first term of an A. P. is 1, and the sum of the first twenty terms is 400; find the thirtieth term. (il6. The sum of the first five terms of an A. P. is one-third the sum of the next five terms; the tenth term is 19; find the sum of n terms. 0-17. Find the sum of eleven terms of the A. P. whose sixth term is 10. 08. Find the sum of the whole series formed by inserting m arithmetical means between a and b. J9. Find the (m+ 1)'" term of , a A. P. whose sum to (£w+ 1) terms is (2»i+l)c. 20. Show that the sum of the r"» term from the beginning and the r«' term from the end of any A. P. is constant for all values of r. , is to the sum 21. The sum of n terms of the series, 1, 4, 7 . of 2n terms as 10:41; find n. ^2. The sum of three terms of an A. P. is 33, and the sum of their squares is 413; find the terms. ^, and - 5. iweer. nr eries ? P. is 30, :erm and fifth by find the the first hird the the sum se sixth rting m (£w+l) gmnmg . for all he sum sum of ARltHMfiTrCAt PftOORfiSSlON. ^5 C^23. The sum of three terms of an A P is 1 5 „„^ +k their cubes is 495; find the terms. ' """ '""^ "^ ^ lusei ted whose sum is greater by unity than their num W ■ how many means are there ? i»»m»^r, now c25. The sum of five terms of an A P is 25 anrl fK. ten terms is 100; find the sum of n tms. ' ' "™ '' <^6. The sum of four numbers in A ia 44 „r,^ +1, • is 13440; find the numbers. ' *^''' P'°^""* 28. Divide unity into four part, in A. P. .ueh that the sum of ^ leir cubes may be --. V rVNs>H^wA»'^ 10 V'v their cubes may be --. lour, but starting two hours after the former- in hr.^r pr;;r :e!r- -- - - - -cr r;- Ct ;*" """ °* " '^""^ O'™-" "y *••<• «-t tenn is a per-' a".^if;L'":~;^oatt:^r "— — -- 33. n 5 be the sun. and ,/ the difference of an A. P. of ,. tenns, ^ 1 r?i 66 mOHER ALGEBRA. then the difference of the squares of the first and last terms is -(n-l)dS. n ^4. The middle term of an arithmetical series of n terms is j!>, and the n^^ term is q times the middle term ; find the first term and the common difference. 35. Given the first term and the common difference, find n so that the sum of 2n, terms may be equal to p times the sum of n terms. Examine the case in which d = 2(i and p = i. 36. The series of natural numbers is arranged in groups thus: 1, 2 + 3, 4 + 6 + 6, etc.; find the first and the last numbers of the the sum of the n^^ group, and the sum of all the n^^ group, groups. -^^ 37. The odd numbei's are arranged in groups thus: 1, 3 + 5, 7 + 9 + 11, etc.; show that the sum of each gro.up is a perfect cube, and the sum of any number of groups beginning with the first is a complete square. /38. Find the sum of n terms of the series, 1 + (2 + 3 + 4) + (5 + 6 + 7 + 8 + 9) + .... 39. In any arithmetical series the sum of any two terms, less the first term, is a term of the series ; and the difference of any two terms, increased by the first term, is also a term of the series. 40. If the terms of an A. P. be arranged in groups of n terms each, the sums of these groups will form an A. P. whoSe common difference is n^ times the common difference of the original series. 41 Prove that the terms of an arithmetical series will still be in A. P. after any of the following operations have been per- formed upon them : (1) If the same quantity be added to or subtracted from each. (2) If the terras of another arithmetical series l)e added or sub- tracted in order (3) If the terms bo multiplied or divided by the same factor. cases. ARITHMETICAL 1>ROG11ESSION. 67 90. The student should carefully work out «.!! fKo a-^ . cases of Art. 88 This will off a if, *^® different . -A .' ^'^^^^^^' ^ome of the results are worthy of sDeeial consideration, of which the following is an example: "^ Oiven /, c?, ^, to find w and «. From the fundamental equations, l='a + {n~\)d ^j^ Substituting in (2) the value of a obtained fn>m (1) we get ^o = n{2l + d-nd), or n '2T • from which Similarly from the same equation, h may be shown that may be fold ^ ^' "'"" ""^'P^^ing values of „ anrr Th!" ''/ """/u^"' «'™" *» ^' '« ™'-s each for n and «. The nature of the problem evidently requires „ to bl T »umiiin;trrl;rdrdmor«'^ ^" ^'■^" '•-^ *- -'- A numerical example will show how this is possible : ^' ?=11, rf=2, ^=32, which give n-4 or 8, and a = 5 or -3. We thus obtain o, 7. two series, 9, 11, and -3.-1 l •atk of which satisfies the required condition! 3, 6, 7, 9, 11. iV iH'"' es HlGHfiR AI^EBM. I i |:il «iii Similarly the double values for n and /, obtained wlien a, d and ^ are given, should be examined. 91. "We have seen (Art. 88) that of the five quantities, a, I, d, n, S_ three must be given in order to determine the remaining two; if, however, the form of the w*'' term, or of the sum of n terms, be given, the whole series is immediately known. We have seen that the n^^ term is a + (n-l)d, and the sum of n terms, ^{2a + {n-l)d}. These may be written {a-d) + dn and \ ~ 2/^"^ \2/^*' '^^^^^ ^^^^ *^** *^® ™ost general forma of these expressions are p + qn and rn + Sn\ where p, q, r and S are constant numbers for any particular series; but n is a variable number, by giving uifferent values to which in succession any particular term, or the sum of any number of terms, may be found. V92. Given the w'* term of an arithmetical series, p + qn, to find tlie common difference and the sum of n terms. The successive terms are formed from the general term by substituting for n the values 1, 2, 3 n in succession. Then S={p + q) + {p + 2q) + {p + 3q) + ....{p + nq) =p+p + ..., to n terms + {1 + 2 + 3....n)q n(n+l)q The common difference is evidently q, the coefficient of n in the general term. / 93. Gitfen the sum of n terms of an arithmetical series, pn + qn\ to find the w'* term and the common difference, Liet aS_. oenofA t.hft snm rwf « fAi>rna *vw j ^*«2 4-1,''— Cf iii -„ _i — .,_ ,_ „..,5i., — j,,i.-^^j^^ tHcn *-J„_i ".Vlli denote the sum of (» - 1) terms =jo(w - l)+^(n- I)*, Now, if ARITHMETICAL PROGRESSION. 69 from the sum of n terms we subtract the sum of n - 1 terms, the remainder must be the w'" term. Then w"'term = AS' -S =p-hq(2n~l). The common difference is 2q, the coefficient of n in the n^ term. The first term may be found by writing 1 for n either in the expression for the n- term or for the sum of n terms. .r^ ^* T'!l ^ ^"^^^'"^^^^^ *« give a different solution to the problems of the two preceding Arts. In every arithmetical .jeries we have w*'*term = a + (ri_l)c/. ^°^'i^ n''^ term =p + qn = (p + q) + (n-l)q, Again, If, then, n S^pn + qn? = ^(2^ + 2^^) = ^{Hp-^q) + {n-\)2q}, the first term is ;, + y, and the common difference 2^, as before. me!nin!t ""'"'." '' '"T ^"'^''"^ ^^*' ^^^^^^^ - *« --gn a meanmg to negative and fraofj^^oi „„i..__ <• , . . . ® JEx. 1, Art 89 s—sonai vaiuca oi « oDtained as in 70 HIGHER ALGEBRA. Let n= -m be a negative value so obtained, then — m written for n satisfies the equation. :. S=^{2a-(m+l)d} m -{-2a + (m + l)4 m ^{2{d-a) + {m-l)d}. This shows that S is the sum of m terms of the series beginning with d — a. Again, ^=_{2a-(m+iy}, m S=-{2{a-d) + (m-l){-d)}. This result shows that if we begin with the term a-d and count backwards m terms the result will be - S. Similarly, meanings^nay be assigned to fractional values of n. ^.iL 96:||^^.— The /" term of an A. P. is P, and the y*** term ,y^lj^ Q; find the {p + qY^ term. We have p^^ term = a + {p-\)d = P, and q^^\^Ym=^a + {q-\)d=^Q. Therefore p-q Now (p + qf' term = a + {p + q-l)d = a + (p-l)d + qd p-q pP-qQ p-9 (vritten ginning d and lilarly, '^ term ABITHMETICAL PROGRESSION. 71 ^..^^.-Find the condition that ., y a^d . may be the p- ^th a,^j} ,,th terms of an A. P. We have y'''term = a + (y_l)^^y^ v,th term = a + (r-l)d=.z. From these three equations we must eliminate a and d. Sub- tractmg the equations from each other in succession we get ■ {p-q)d^x-y^ {q-Ad = y~z, {r-p)d=z-x. Now multiply the equations respectively by r v and . «.n^ add. then the leffside vanishes and we get ^' ^ {^-y)r-\-i,j-z)p + {z-x)q=.(i^ the condition required. Had we multiplied by ., . and y instead of r, ;, and .7 we should have obtained « y instead the same result under a different form. -^n+a -^n = (w + !)"> + (w + 2)'» + (n + 3)th terms = 3(w + 2)"'term which proves the proposition. ing ;, te™, "t: ''™^' """^ "'^ ^l"*' *" *"« -» »« *"« follow- {•rn _i_ /v. \ „ / - - - \ {n+p) m{n~pY 11 I f ft 'Q 72 HIGHER ALGEBRA. Equating the sums of the three sets of terms we get ^{2a + (m- l)d} = I {2(a + md) + (n- l)d} Therefore and ' X From which and Then by division = ^{2{a + m + n.d) + (p-l)d}. V. 2a + (7n-l)d ^n 2{a + md) + (n - l)d~ m 2(a + ni + n.d) + {p-l)d n 2ia-pmd) + {n-\)d ^p ' ^^ ^/ > \j ^ {vi + n)d \ in- n 2{a + md) + {n-\)d m (u +p)d n ~ p 2(a + md) + (n-l)d~ p ' m, + n p (7n - w) n +p 7n{n — p)' EXERCISE VII. / 1. How many terms of the series, 25, 23, 21 , must be taken that the sura may be 165 1 J^. The n^^ term of an A. P. is 2w- 3; find the common differ- ,4nce and the sum of n terms. A* C3. The r^^ term of an A. P. is 7 - - ; fir^/' the sum of 2%+ 1 ^terras. / 4. The {n-\.\Y^ term of an A. P. is ^^^f^^ find the sum of / 2n+l terms. , Q 5. The n^^ term of a series is w'- n+ 1 ; write down the tenth |, term, the r**> term, the (w + 1)*" term. Is it an arithmetical ■ §eries *? c_ 6. The sura of n terms of an A. P. is dn^-Sn, find the n*" ^ (a 'j term and the common difference. 1^^ ARITHMETICAL PROGRESSION. 78 i)d}. , must be Qon differ- of 2n+l he sum of the tenth ithmetical id the n*'* . \j J. The sum of n terms of the series, 2, 6, 8 . . . . , is 950; find n ^ Give an mterpretation to the negative value of n. c3. Find the eighth term of an A. P. whose sum to n terms is ^'^ 2l3'*'4j* 9. The nt" term of an A. F.h2n+1; of how many terms is the sum 99 ? Give two different interpretations to the negative result/. ^10. If the sum of p terms of an A. P. equals the sum of a terms, then the sum of jp + ^ terms is zero. ^11. Find the relation between a and d when the equation be- tween a d, s, n, is satisfied by two different positive integral values of w. ^6 Cl2. The sum of n terms of an A. P. is an?+bn, and the sum '' of m terms of another series is hm' + am; the fifth term of the first series is half the fifth term of the second series; show that 17a = 7o. 13. The sums of two arithmetical series, each to n terms, are to each other as 13 - 7^^ to 1 +3n; find the ratios of their first terms, their second terms, and their r*"* terms. 14. If a, b and c are the y^ q^^ and r'^ terms of an A P then p q and r are the a^\ 6* and c'^ terms of another A. p' a, 6 and c being positive integers. ' (1^6. If the (p + qyr. ^^,j ^^__^y, ^^^^ ^^ ^^ ^ ^ ^ ^^ ^^^ ^. n respectively, find the p'^ and q'^ terms. ^ 16 T)' 'iJ w(w + 3) ^ . i^ivide -— jg— into n parts such that each shall exceed the preceding by a fixed quantity. , ^17 The sum of m terms of an A P. is n, and the sum of n terms is m; show that the sum ot m + n terms is - (m + n), and th ~ ' " ^ . . . f 9»,\ n terms is (?«. - w) ( 1 + IT ) I ^ f 74 HIGHER ALGEBRA. Q 18. If S^ denote the sum to n terms of an A. P., show that "n+2-^S„+i + Sn is equal .^ v,. v.....siuon diflference. 19. If .S'„ denote the sum of n terms of the natural numbers beginning with a, pi ove >S'3^+„_, = 3;^^. 20. If a,, flj, ttg a^ denote the terms of an A. P., and S their sum what is the sum of the series, 21. There are n arithmetical series of n terms each; their first terms are the natural numbers, 1, 2, 3 . . . . , and the common difference of each is the same as the first term- find the sum of all the terms of the series taken together. f 22. If ,S' and .S" denote the sum to n terms of two arithmetical series having the same first term a, and common differences d and -(i respectively, then ^^^^fezil^. 23. If S„ denote the sum of n terms of an arithmetical series, and (^S,„-3S,, + 3S, = 0. 24. If the sum of the first m terms of an A. P. equals the sum of n terms beginning with the {r+iy\ and also equals the sum of ^ terms beginning with the (s+ 1)*", prove 2r+n-m (m~n)p '28+p-m {m~p)n ^ 25. If the y" and q'^ terms of an A. P. are P and Q respec- tively, what is the n^^ term ? V 26. If S be the sum of n terms of an A. P., and S' the sum of ihe arithmetical means between the consecutive terms, then aS': aS" = w: w— 1. nS £ ^1. If «: + aa -f-flg + . . . . a^ = ^^ gho^ ^j^at (S-a,y + (S-a.y + ^. ff 2 + . . . . a,j". p., show that ural numbers A. P., and S )'i !h; their first the common d the sum of arithmetical 5rences d and letical series, aals the sum lals the sum id Q respec- ' the sum of IS, then ARITHMETICAL PROGRESSION. 75 .K?:J^^ ''"*'' ""^ * right-angled triangle are in A. P.- show that they are proportional to 3, 4, 6. ' ^29. If a\ h\ c- are in A. P., then -L J_ ^ . , in A. P. 6 + c'c + a' ^6 -«-!-> m A. P.; also, a^ - be, h^ - ca, c^ - a6 are in A P. Compare th^ common differences in the several cases. ^ "^ C rr^b' *' FTTc ^^^ ^^ ^^- P-> then A, *, 1 ^^e in A. P. 32 Tf ''^ i c b-c' 7:ra^ iTTi *^« in A. p., then aH^-2A3 a + 6 + c — 2"— • a' + c2-2A2' (,33 If two terms of one A. P. are proportional to the corre- sponding terms of another A. P., then all the terms of the foZ senes are proportional to the corresponding terms of the Lter 34. If ^„ denote the sum of n terms of an A. P., then r S^+2p - 2S,,^p + S^ =fd, ^AZttr^' ^' '''^"""^ "^'^ ""^^^ "^ -^-h the sum of the first half of any even number of terms bears a constant ratt r ler '' ''' ''-'''' '''' -' «^°- *^- *^ere is butt: diieL'trr2T'^^l:n '^'^ ^^ ^- ^-^ -'-'^ — *,2. »K. .u : ., . ' *""* "^^^^^ s"™s. each to n terms are « , show that the.r fi.t terms f„™, . decreasing A. P. wC first term is g („ + 1), ^„<, „„„„„„ ^jg.^^^^^^^ 1 ^^ _ 2 t/37. If ^, ^V ... be the sums of .n arithmetical series, each 76 HIGHER ALGEBRA. to^ terms, the first terms being 1, 2, 3. . . . , and the differences 1, 3, 6 .... ; show that S, + S, + . . . . S^= J wn(mn + 1). , ^,38. If Si denote the sum of n terms of the series 1, 6, 9 and S, the sum of w - 1 terms, or of n terms, of the series 3, 7* 11 ... . , then Si + S, = (Si-S,y. ^ f 39. Find the first of n consecutive odd numbers whose sum is on", where p is any positive integer greater than unity. 40. Show that nP may be resolved into the difference of two -.integral squares, n and p being integral, and p greater than 2. 41. If S, denote the sum of n terms of the natural numbers beginning with r, then S^^S^^ . . . ,S^ = ni n{m + n) ^ < 2 \,42. The successive terms of an A. P. are arranged in groups of 1, 2, 3, etc., terms each; show that if S^ denote the r*"" group, then and n Sn = na+-{'n?-\)d^ '^i + /^2 + ....Ay„ = ^^^^~J{4a + (n-l)(w + 2)c/}. 43. With the notation of the preceding example show that and p{S,, - S,) + m(S„ - S^) + n{S, - S^) = (m - n){n - p){p - m){m + n+p)^. ''iSc \'-if (»•),; ;^ CHAPTEB VI. cut"' _^ ral numbers GEOMETRICAL PROGRESSION. 97. A Geometrical Series, or a Geometrical Progres- sion, IS a succession of numbers which constantly increase or constantly decrease by a common factor. Thus each of the following series is a geometrical progression: 1, 2, 4, 8, 16 .... 72, -36, 18, -9, 4^.... a, ar, a«^ terra and th common ratio being given. ^"^"''^^' Forming the terms in succession we have v 1«' 2nd 3rd joth ^t^ «. «^, ar2 .... ar^ .... a^-i^ from which we observe that— Any term is found by multiplying the first term by the common factor ra^sed to a p^er less by one than the number of the tZ This result is briefly expressed in symbols thus : w"' term = ar"-^ 101. To find the sum of a given number of terms of a aeo metr^cal ser^es, the first term and the common ratio I fg uZ^ ^^Let . be the number of terms required, and denote the sum Then 'S'=a + ar + ar' + ar3 + ....a^n-2 + ^^n.i therefore rS^ <^r + ar^ + ar> + .. , ,ar^-^ + ar^-r^ar\ Then by subtraction rS-S=- ar^-a «(r»-l) "TTT- (2) rl = ar\ substituting in (2) we get S== '^-tll r~\' (3) Equations (1), (2) and (3) should be .:on,n,itted to memorjr. therefore Again, from (1) *s'=: 102. If any two terms of a G P be mv«.T, +»,-> • • P.t.. dete^in.. .0. suppose the ^r^: ^ ~ then and .elT" tZ,T ^""""""^ " ""^ - ™'^ '"' '™"''- »<• *»- the ay."*-! -,p ar**-^ = g. GEOMETRICAL PROGRESSION. 79 103. To find the geometric mean between two given extremes. Let a and h denote the given extremes and x the required mean, so that a, a*, h are in G. P. Then from which a X x= Vab. From the above it is seen that the geometric mean between two quantities is > The four cases in which n ha^ to be found require logarithms for their solution. The rema,ining cases, viz. : when (1) «, n, s, (2) n, /, «. are given, are incapable of a general solution. EXERCISE Vni. Find the required terms in the following series: 1. The fifth term of 2, 4, 8, 16 2. The tenth term of 1, 3, 9, 27 3. The fifteenth term of 64, 32. 16 8 4. The w**" term of 2, 6, 18, 54 ... . 5. The {2n - !)»•> term of 4, - 8, 16, - 32 ... . 6. The v}^ term of a, «V, aV, aV Q* 7. The 2n*'> term of -, - a, A, - - . . . . 8. The (n - 1)'" term of 3a, 5aV, 7a^r% 9aV. ^^'S V3 3 - V 3" 9. The w*** term of V3 + l' 1/3+2' ^^3 + 2 10. The(2w + 3)tHerm of ^±1 1 1 V2-I' 2-7f' 2 N^ ^N on. When y theoretic- ipt to work at some of ved by the i^ed by any . (2) a, n, I, logarithms are given, f^N GEOMETRICAL PBOORESSION. §1 li. Sum 1 + 2 + 4 + 8. ... to ten terms and to n terms. 19 Q 8 8 40 iw. Hum -+-+-_ to SIX terms and to n terms. 13. Sum 1-2 + 4-8.... to 2w terms and to 2n+l terms. 14. Hum 2 + - + - + __ to ten terms and to n terms. 15. Snma^x-^-a^+a\r-a^j;\... to n terms and to 2ri- 3 terms. / 16. Sum (2- >/3-) + l+(2+ ^3). . . . to n+ 1 terms. 17. Sum to n terms the series whose ri*"* term is ar^-\ 18. Sum to w+ 3 terms the series whose n^^ term is a^-%'^-\ 113 Cl9. Sum - + -+-+.... Tto w+ 1 terms. 20. Sum 2 + ^8 + ^ 2 + . . . . to w terms and to 16 terms. 91 o 5 5 10 ^1. Hum ---+__.._ to n terms and to 2w+ 1 terms. 22. The first term of a geometrical series is 5 and the third term is 80; find the common ratio. x' 23. The fifth term of a geometrical seri'js is 48, and the ratio is 2; find the first and n}^ terms. 24. If a = 2 and r = 3, which term will be 162 ? 25. The sum of three terms of a geometrical series is HI, and their product is 8; find the t^rms. ^6. The sum of three number^ Lx G. P. is 13, and the sum of their squares is 91 ; find the numbers. 27. The sum of two r imbers is m, and their geornetric mean IS n; find the number;. 28. Insert two geometrical means between 9 and 21|. 29. Insert six geometrical means between /^ and 5^^. 30. What is the common latio of a geometrical series when 82 HIGHER ALGEBRA. IS; h tJie difference between the first and n'^ terms is equal to the sum of w - 1 terms 1 y31. The sum of the first three terms of a geometrical series is 4f , and the sum of the first, third and fifth terms is 8-^'^; find the series. v32. The sum of the first six terms of a geometrical series is 157^, anrl the sum of the third to the eighth inclusive is 630; find the series. ^33. The fir^t of four numl,ers in G. P. is ^, and their sum is greater by one than the common ratio; find the numbers. 34. There are five numbers, the first three of which are in G. P., and the last three in A. P., the second number being the common difference of these three terms. The sum of the last four is 40, and the product of the second and last is 64; find the numbers. ^35. Three numbers whose sum is 27 are in A. P.; if 1, 3 and Tl be added to them respectively the results will be in G. P.; find the numbers. 36. To each of the first two of the numbers, 3, 35, 190, 990, is added w, and to each of the last two is added y; the resulting numbers are in G. P.; find x and y. 37. The series of natural numbers are divided into groups thus: 1, 2 + 3, 4 + 5 + 6 + 7, etc., each group containing twice as many numbers as the preceding; find the sum of the «*" group and the sum of all the groups. 38. The odd numbers are divided into groups thus: 1, 3 + 5 7 + 9 + 11 + 13, etc., each group containing twice as many num- bers as the preceding; find the last number of the w*" group, the sflm of the w"' group, and the sum of all the groups. 39. The terms of the geometrical series, 1, 2, 4, 8 , are arranged in groups thus: 1, 2 + 4, 8+ 16 + 32, etc., each group GEOMETRICAL PROGRESSION. 83 lual to the al series is i 8-1% ; find *1 series is ^ve is 630; leir sum is 3rs. ich are in being the f the last ; find the f 1, 3 and >. P.; find )0, 990, is resulting io groups f twice as n*** group 1, 3 + 5, iny num- roup, the ch . , are group containing one number more than the preceding; find the sum of the n^^ group and the sum of n groups. 40. The terms of the geometrical series, 1, 2, 4, 8 are arranged in groups thus: 1, 2 + 4, 8 + 16 + 32 + 64, etJ.,* 'each group containing twice as many terms as the preceding; find the sum of all the groups and the sum of n^^ group. 41. If a geometric progression consist of in terms, show that the ratio of the sum of the first n terms and the last n terms is to the sum of the remaining 2n terms as r^" _ r" + 1 to r". 42. Find the sum of the squares of the differences of every two consecutive terms in a G. P. of n + 1 terms. ;43. Determine m and n in te)'ms of a and h so that ^^ + ^^ may be the a-ithmetical mean between m and n, and tC gZ- metrical mean between a and b. 44. In a G. P. with the usual notation prove «^2n = S,(S,,^, - r . S„_,) and ^..(^3„ - ^,„) = (S,, _ s.f, 45. If P bo the continued product of n quantities in G. P., S their sum, and S' the sum of their reciprocals, show that 4'6. If S,, denotes the sum of n terms of a G. P., S^^ the sum of the next 2n, terms, and ^%„ the sum of the following 3n terms 47. If K + a,' + a,^ + .... «„.,=')« + ag^ + a,2 + . . . . a J) = (aia2 + a2a3.+ ....«„_ia„)2, and ai, aj . . . . a^ are all real, then a^, a^, . . . . a„ are in G. P. ,. ^^^«f r'»^!.Tr °^ ^ ^^^^ ^^ * geometrical series has been showK to be ^^^^; this may be written "^^^ Now if r be less than unity, by taking n large enough »-« may he made as Hrxi^li IB I 84 BIOHER ALGEBRA. as we please (Art. 15), i.e., the sum of n terms may be made as nearly equal as we pleaae to ^. This statement is usually abbreviated thus: The sum of the series, a + ar + ar' + . . .. ad infinitum, is -^. 1 —7* The same result may be obtained thus: then rS== ar^-ar^Jra-fi^..., .'. S(l-r) = a, or S=-^ as before. 1 -r This process deserves very careful attention. Since the series m each line is continued indefinitely it is assumed that the terms cancel each other entirely. In reality, however, am term is neglected, which corresponds with assuming r« to be zero in the former investigation; this is legitimate only when the term omitted is indefinitely small. The necessity for care in such matters will easily be seen from the following: Let ,S'=l + 2 + 4 + 8 + .,.. ^^^» 2^= 2 + 4 + 8 + .... This absurdity arises from the fact that the single term neglected IS more important than the whole of tliose retained. 107. Recurring decimals in arithmetic are familiar examples of infinite geometrical series. Thus -7^-777. ...=.1 + 1(1]+1(1\\ lo^iovio/^ioVio/ ■*"•••• which is an infinite geometrical series, whose first term is — and common ratio — ; its sum to infinity is therefore 10 ■ V 10/ ~ 9 • 10 GEOMETRICAL PROGRESSION. 85 Again, .234^-^4 34 34 34 10 2 10^ 34 10« 10^ 10 10^ {1 + = T?C-t 34 10^ 1 1 10^ + } 10 lO^' =_?. ^* 1- 10» 10 990 232 ~990* The value of any recurring decimal may be found in the same way, but the result may be more neatly obtained by the method of Alt. 106; the solution is given below. 108. To find the value of a recurring decimal. Let r denote the figures which do not recur, and let them be -p in number; let Q denote the recurring period consisting of q digits; let D denote the value of the whole decimal. Then Therefore and Therefore or I>= 'PQQQ.... l0^xD= F-QQQQ.... 10^+^ xD = PQ-QQQQ.... (10i'+«- 10^)2) = P^-P, PQ-1' D = (10«-1)10^* Now, 10«- 1 is a number consisting of q nines, and 10^ is a unit followed by p ciphers. From this result the truth of the ordinary rule for reducing a recurring decimal to its equivalent vulgar is at once evident. ^^ ' HIGHER ALGEBRA. 109. To find the value of nr- when r is less than unity and n 18 indehnitely great. ^ Since r is less than iinifv 1 « ;„ ^ •,.' -, ,no,, 1*1 , ^' 1 - »• 18 positive, and a number, ar, may bo taken large enough so that r X Put then < 1-r, or n +-L< 2. l + -j»* = wi, sothatmve series is sometimes called an arithmetico-geometric series. The last term may be written «r»-i + d{n - l)r~-i. If r be less than unity, by taking n large enough this term may be made mdehnitely small (Art. 109), and may therefore be neglected Omittmg this term and summing the series dr + dr'' + . . . to infinity we get -^ + -_^^ as the sum of the preceding series to infinity. EXERCISE IX, Sum the following series ad injinituni: ,111 o 1 1 1 3 9 27^ 5. 2 1 3 3 2"^8~""' 7. 2-^4+^2-.... 9. 1 1 a/¥ 2 V'2 3 "*" 9 27'^ 11. V2+1 1 1 V2-I' 2-V'2' 2'" 13. Sum l + 2.c+3a:H4a:3^^ less than unity. 2 ^ ^u.1 1 2 4 8 16 .333 i ~ 8 "^ l6 " • • • • 6. 2+\/2 + l+.... 8. (2+V'3")+l + (2- v/3)... 10. a+/_j L Sb b 12. ^^^ __y±_ 3-V3 '/3'+l' v/3 + 2' V'3 + . to w terms and ad inf., a^feng 88 HIGHER ALGEBRA. C^ A A J 14. Sum 1 + 0+92+03 + '*" t®** terms and art iryf. (D15. 3 5 7 Sum l + ^+o2 + S3 + "" **'** terms and ad ir\f. 16. 3 5 7 Sum 1-^ +92~9i + *'" ^** terms and ad inf. \ /^ 17. Sum ar+ {a + ah)r^ ■\- {a-{-ab + ah'^)t^ + . . . . to n terms and ad inf.y r and hr being each less than unity. ^18. In a geometrical series consisting of an odd number of terms prove that the product of the first and last equals the square of the middle term. ^19. The (n+Vf^ term of a geometrical series is c; find the product of 2n + 1 terms. /T 20. If n geometrical means be inserted between a and 6, what is the product of the terms of the whole series thus formed 1 '^ 21. In a G. P. the {p + qY^ term is m, and the {p — qf^ term ^s n'y find thejp*'' and the 5'*'' terms. 22. If the p^'^ term in a G. P. is P, and the q^^ term is Q^ what^ is the w"" term ? term is ^, what/ of a geortfetrical 23. If a, h and c be the jo***, q^^ and r^^ terms progression, then a*"'', y-^. c^~^= 1. 24. If the j»*^ q^\ r^^ and s* terms of an A. P. are in G. P., then p - q, q — r and r - « are also in G. P. c25. In a G. P., if each term be added to, or subtracted from, the preceding, the results in either case will be in G. P. 26. If the terms of a geometrical series be arranged in groups of p terms each, the sums of the successive groups will be in G. P. Find the sum of n groups, and show that it is equal to the sum of the same terms taken separately. .^ 27. If S be the sum of an odd number of terms in G. P., and is c; find the )tracted from, GKOMETRICAL PROGRESSION. 89 if .S" bo tho sum of the series when the signs of the even terms are changed, sliow that the sum of the squares of the terms will be jSij . 28. If there be any number of quantities in G. P., r the com men ratio and .S„ the sum of the first n terms, prove that the sum of tho products of every two terms is r+ 1 •^"•'^n-i' C25- If P be the sum of the series, 1 +rP + r'p + r^p+ „,i ,•„/• and if^ be the sum of the series, H-»'«+ r'* + ,*/ + ... ac/ inf show that P9(^Q -\)p = Qp/p _ j \ ?_ -^ "' ^30. Sum the series, V'^+ | V 3 + ? v' 2"+ . . . . «e/ ir^. k_- 31. Sum the series, ' ^. J V' 2(1 + V 2") " (r772)(2T71) ■*■ ^^^^ C 32. Sum ton terms 3 + 33 + 333 + .... ,^.g, lU-'^A 33 Find the series in which each term equals n times the sum of all which follow it, the sum of the first two terms being m. C 34. If S, be the sum of n terms of a G. P., what are the sums of/Sfi + ^2 + ....^ and ;S^„ + ^^ +^ + o 35. Show that 2^4^ 8* . U''' . . . . ad inf. == 4, and that i4 u^.^ 3^9^.27''^81''^ . ad inf, = 3 . L- • '^i, ^ ^3 .... ^„ be the sums of n terms of n geometrical progressions, of which all the first terms are 1 and confmon ratios 1. -, ^ . . . . n respectively, show that '^i + .^. + 2.% + 3^, + ....(^_l)^^^l„^.2n + 3„^__^^^^„ ^ C37 If ^„ denote the sum of the n"' powers of the terms of an infinite geometrical series, show that 1^ 1 1 1 1 r a —r' /.- ,.;-■ IMAGE EVALUATION TEST TARGET (MT-3) A // ^•^. *>,*- * •■*■"...; <- 1.0 I.I 1^128 |50 "■■» - lis 11 10 m 2.2 IL25 III 1.4 — A" 18 1.6 FiiotDgraphic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y, 14S80 (716) 872-4503 ^ % i\ '^ '"^^T^ ^""^^ "^"^ 4 ^, 4^ i/.. :A ■ I ^ 90 HIGHER ALGEBRA. 38. Given S the sum and s^ the sum of the squares of the f^ terms of an infinite G. P terms is S show that the sum of the series to n {-G^5"}- (^ 39. The middle points of the sides of a triangle are joinod, the middle points of the sides of the triangle so formed are again jomed, and so on ad infinitum. Show that the sum of the areas fj'r all the triangles so formed is one-third the area of the original triangle. W 40. Two straight lines meet forming an acute angle; from any \joint in one a perpendicular is drawn to the other; from the foot of this perpendicular a perpendicular is drawn to the former, and so on ad infinitum. The lengths of the first two perpendiculars are a and h ; find the sum of the lengths of all the perpendiculars and the sum of the areas of all the right-angled triangles thus formed. 41. The triangle ABG has each of the angles at B and C double the angle at A ; lines are drawn within the triangle, making the triangles CDB^ DEB^ etc., each similar to the original triangle. If A denote the area of the original triangle, find the sum of the areas of the infinite series of triangles, ABC, CDB, DEB^ etc., and also of the infinite series, CDA, DECy etc. iquarea of the .he series to n are joinod, the ned are again m of the aresis of the original igle; from any ; from the foot he former, and perpendiculars perpendiculars triangles thus 5 and C double le, making the ginal triangle, the sum of the »^, DEB, etc.. CHAPTER VII. HARMONICAL PROGRESSION. Ol^' •^'" "^""^°"^^^^ Series, or an Harmonical Progres- Sion, IS a senes of numbers such that of every three con3!ve ^rms the ratio of the first to the third equal the Vatof « difference between the first and second to the difference between th„d and third, the differences being always taken inTh: Thus a, b,c,d.... are in harmonical progression b:d=b~c:c-dj etc The numbers 30, 20. 15 12 10 o-o • i. gression, ' ^^ 10.... are m harmonical pro- ^^'^ 30:15 = 30-20:20-15, 20:12 = 20-15:15-12,'etc. thetttirH. R ^^"^"'^^' ''''''''''''''" -^ ^^-«^ ^-«^^ ^y fgr^^o^ZT"^ Progression, formerly called Musical Pro- Ipnrr^l, ''"'f''^^'^ ^""^ ^^nsion produce harmony when their engths are in progression according to the preceding definit on Is importance is chiefly due to this fact and to the occur ence of Wonical quantities iu connection with many ;Z::Z w HIGHER ALGEBRA. 113. A good example of quantities in H. P. may be obtained as follows : Take any two straight lines AB and CD cutting each other in C; bisect the angles ACD, BCD by g the straight lines CE, CF; across the C four lines CA, CE, CD, CFdYa.\v any straight line AEDF, then the lengths of the lines AE, AD, AF are in har- ^ monical progression. A E F For AE:ED = AC'.CD Euc. VI., 3 = AF:FD. Euc. VI., A Therefore AE:AF=ED:DF Art. 52, (2) = AD-AE:AF-AD, which shows that AE, AD, AF satisfy the conditions for H. P. according to the definition. 114. If a series of numbers are in H. P., tlieir reciprocals a/re n A. P., and conversely. Let a, h, c be in H. P. Then, by definition, a:c = a — bib — c. Therefore a(b-c) = (a- b)c. 1_1_1_1 c b b a' Art. Ill Art 48 Dividing by abc, Therefore 1 1 1 a' A' c the converse. are in A. P. The process reversed proves * Cor. 1. — The general expression for an harmonical series is obtained by taking the reciprocals of the successive terms of an arithmetical series. T,, 1 J_ 1 1 a a + d a + 2d a + (n-l)d is a general expression for any harmonical series. HARMONICAL I'ROORfiSSION. ds ay be obtained reciprocals are Cor. 2 A constant quantity divided by the successive terms of an A. P. g^ves^quotientsju H. P. For the reciprocal of. {a + {n-l)d} ^^ a'^^'^'^^c' ^^^ general term of an A. P. wliose first term is '' md ccanmon difference - ^ c' 115. Harmonical progression, though connected with arith- me -1 p"ogress.oa by the simple relation given in the former Art IS nevertheless, essentially different, in several respects from both it and geometrical progression ^ ' The various terms in A. P. and in G. P. can be written when the first term and the common difference, or the common rat.o are g.ven; but m II. P. there is no quantity corresponding S the common difference or the common ratio. Again, in the L. progresszons convenient expressions can be found which repre' sent the sum of any number of terms; but no correspondin/ex- ^r^Z:^-"' ''''-' '' "^^"^^ ''^ ---P^-^-^ arith- 116. If two terms of an harmonical progression be given, the series IS completely determined. « given, the For let tlie m^^ and n'^ terms be j. aiid g. L'eversea proves ■ ■ Then H and ive terms of an 1 ■ !!« 94 HIGHER ALGEBRA. 117. Tojind the harmonic mean between two given extv.mea. Let a and b denote the given extremes and x the required mean, so that a, a*, b are in H. P. Then Therefore from which a:b = a — x:x — b. a(x — b) = b(a — x)y 2ab a; =3 a + b' Thus the harmonic mean between two quantities is twice their product divided by their sum,. 118. To insert a given number of Jia/rmonic m^ans between two given extremes. Let a and b be the given extremes and m the number of means. Insert in arithmetic means between - and - ; their reciprocals a b will be the harmonic means required. The arithmetic means are : 1 1 /1 1\ 1 _2_/l_l\ 1 7n /I l\ a m+l\b ar a m,+ \\b a/"" a m+l\6 a/* Simplifying, and taking the reciprocals, we get {m,+ \)ab {m+V)ab {m+\)ab mb+a ' (m-l)b + 2a' '" b+m^ ' the harmonic means required. This series can easily be remembered by observing that the numerators are all alike, and that the denominators are the same as the numerators of the corresponding arithmetic means taken in the reverse order. Cor. — The product of the r* arithmetic and the (m-r+l)^ harmonic means between any two quantities is equal to the square of the geometric mean between the same two quantities. 5S is ttoice their .713 between two ■ /J HAKMONICAL PROGRESSION. 95 -119. If A, G ■ X V z y are in A. P. Add a unit to each, then Therefore a a a - , - , - are in A. P. X y z X, y, z are in H. P. f Art. 114, 1 Cor. 2. i, c are also P. EXERCISE X. 1. Find the tenth term of the series, 3, 4, 6 ... . 2. Find the w*'» term of the series, - , - 1 ' 3' 6' 3. Find the twenty-fourth term of the series, 24, 12, 8 ... . 4. Find the arithmetic, the geometric and the harmonic means between 2 and 32. 5. Insert two harmonic means between 1 and 2. 6. Insert three liarmonic means between 16 and 4. 7. The second and fourth terms of an H. P. are - and - - • find the first, third, fifth and n''' terms. ^ ^ ' 8. The first and second terms of an H. P. are « and b; find the w"* term. 9. The arithmeti c and g eometric means between two quanti- ties are o - 6 and Va' - b'^ respectively; find the harmonic mea^/' I i^' ; ' ! 08 BIGHER ALGEBRA. 10. The arithmetio and harmonic means between two numbera are 2 and 1 J respectively; find the numbers. 11. The sum and difference of the arithmeti*^ and geometric means between two numbers are 16 and 4 respectively; find the harmonic mean. 12. Find a number such that the arithmetic? mf;an between it and 2 may be 2? times the harmonic mean. 13. Find two numbers whose difierence is 16^, and the geo- metric mean between the arithmetio and harmonic roeans of which is 9. \rU. The sum of three terms of an H. T. ia ~, and the first 1 ^2 term is - ; find the series and continue it two terms each way. tIS. The arithmetic mean between two numbers exceeds the geometric mean by 13, and the geometric mean exceeds the har- monic mean by 1 2 ; find the numbers. j^l6. From each of three quantities in H. P. what quantity must be taken away so that the remainders may be in G. P. 1 yll. The sum of three numbers in H. P. is 11, and the sum of their squares is 49 ; find the numbers. ^8. Find the value of "^ ^t ^^^n a, h, c are (1) in A. P., (2) in G. P., and (3) in H. P. ^19. If «, 5, c are in A. P., and a, mh, c in G. P., prove that a, m^h, c are in H. P. A. 20. If X is the harmonic mean between m and »», show that 1 1 1 1 .= _ + _. X— m X ~ n m n 21. If four quantities are proportionals, and the first three are in A. P., prove that the last three are in H. P. 22. If H be the harmonic mean between a and i, prove that it is also the harmonic mean between H -a and If~h. 1^1 wpi HARMONICAL PROGRESSION. two numbers id geometric ely; find the n between it a,nd the geo- ic raeans of md the first each way. exceeds the jeds the har- lantity must P.? [ the sum of 1) in A.P., , prove that how that :st three are prove that b. r23. If a, 6, o be in H. p., b+a b+c then and are in A. P. u C. 26. If a, 26 and c be in H. P., then will a + c, a and a - i be in G. P.; so also will c + a^ o and c - i be in G. P. 27. Find the condition that a, A, c may be the p>-\ q^^ and r^ terms of an harmonic progression. 28. If the (p + qY'' term of an harmonic progression be m and the (p-qY'' term be n, find the/" and ^"^ terms. f 29. If the p^ term of an harmonic progression be P, and the o*"* term be Q, the w*" term will be {p-q)PQ in-q)Q~{n-p)P' V.30. If a, i, c be in A. P., and a, 5, d in H. P., then • c_ 2( a-&)^ (^ a6 ex. If ^, G?, ^ be the arithmetic, geometric and harmonic means between a and i, then g-1 AS~a){H-h) 1 11 i< ^ ?« and - — - A . = _ ^ GP A + G^G+H G' (> 32. If a:r=.by = cz, and a, J, c are in A. P., then .r, g/, « are in MFT wmmm irV) limnKR ALORBRA. ^IVX U a-«Ai'.-o«, and «, A, „ aro in i). P., then a", y, « are in 34. If oitlmr of two oonsooutivo t«^rmR of an IT. P. is divisif>lo by t|H>ii- (lUroitMico, show that ono of tlio tornm of tho sorioa i« iurinity. Can any tonn of u tinito hannonioal mm-wh Im nm>1 an. If an arithuiotical and an Imrmonical progression have „uoh tho sanio lirst and sm^ond trrn.s, « and A, and if .r and y lie the n"' tonna in tho two Horios, then y(^ - n) n I «(y-/i) n-2* 136. If Iwtwoon any two quantities there are inserteil two arith- metic nuuMis, A„ A,, two goonjotrio means, G^, 6?,, two harmonic means, Jf^, ]f^^ then A^ + A^ I a^a^ , ^ C^l, If (I, A, are in IF. P., then - J? . * ^ are also in H. P. /> + o-2« o + «-26' « + 6- 2c 111111 -, - + CO « + 6 ^38, If «, A, c are in II. P., then i + — . 1 + J:. «»»;»TT -D J -•* . ^ . . ca nh are in H. P., then areinH.P.;.mdifa + Ji-, A + -f!L c + — A + c' o + tt* a + 6 «, 0, c are in A. P. ^^, If (», A, fl Iks *hroe quantities such tlmt a is tho arithmetic moan In^lNvtHMi b and r, and (> is tlio Imrmonio njean between a nnd A, tlu>n «, b, e ai-e in (',. P. Q;lO. If tho luirjuonio moans botween eaoh pair of tho three quantitios, „, A, o be in A. P., then b\ a\ o' shall be in H. P.; but if the Juvrmonio me»ins l)o in IF. P., then A, «, o slmll l,e in H. P. 44. If aS',, iS'a, >/'n. Tho first is ol)- tmuvii hy 8ubtnu,ti(>n; tho sooond is obtained f.-oni tho first bv t'hai 1 »i{in^' n mio n - lb «"oji; tholinul o(jualitio8aroobt.y be applied to find the sum of the fourth, lifth, etc., powers iu succession. 124. Tho sum of tho cubes of tho natural numlx^rs n.ay easily bo found independently of the sum of their squares as follows: Arrange the odd numbers in groups thus: l+(3 + 5) + (7 + 9 + ll) + (13+15 + 17 + 19) + .... In n groups there aro 1 + 2 + 3 . *> = ^ll** "*■ ^ ) 11 — terms. The hust torn, of the n"' group is .r + ,._ 1, the first term is w^-n+ 1 and tliero are n terms in it; therefore the sum of the r.«' group is la This shows that the successive groups are the cubes of the giving !iii-- - tern,s of the series of odd numbers, whose sum is at onco known to be |!i^-'ti)V'' 125. It is frequently convenient to indicate by a single symbol that tuo whole of a series of terms is to bo taken. This is gener- ally done by writing the Greek letter 2' before the ,.»" term of the series, thus: 1 + 2 + 3 + .. ..u is denoted by 2m. 1-' 4.02 , 'VJ ■ 2 •' I + w +0 + n' u ti (' + ar + ar-....ar"-^ « u .Jl^ujn !^""' '"' ^''*"' '""'^"^ *''" ^'«^" ^>^ summation, 0. e must bo t.vken to correctly .distinguish the rarM and rj- ^ t V • t 'f ^'T"^P^^ - - *^« variable; if . were taken for the ^ ariablo, lar'^-^ would stand for a + 2'-»a + r-^« The context will u^n^U.r j.^h- -» • ■ • - - ' nhio , ^ 1 y ^ "" ■^"'' ^'"^^^ "^ doubtful case§ the vari- ftble must be specifietl. ti 2>„.»-i I I \ 104 HIGHER ALGEBRA. Bx. — Sum to n terms the series, 1.2.3 + 2.3.4 + 3.4.5 + n(n+l)(n + 2) The »*"• term may be written n^ + 3n^ + 2n /. sum of n terms = In^ + 3In^ + 2Zn --' ' " 5' /T-i-^X _/ n(w+l) y + Sn(n+l )(2n + 1 ) 27i(»t + 1 ) 6 2 n(n+l)(n + 2) (n + 3) 4 ' 126. When the terms of a series are alternately positive and negative it is sometimes necessary to consider separately the cases in which n is even or odd. The two results can then be combined as follows : Let A and B denote any two quantities. Then ^ +(- l)"if denotes their sitm when n is even, but their difference when n is odd. Let p and q denote the sums of the series when n is even and odd respectively. Then let A+£=p and A-B = q, from which ^A{p+q\ B=l(p-q). Therefore -{(p+^) + (_l)"(^_y)| is the required sum whether w be even or odd. ^a;.— Sum 1-2 + 3-4 + 5-.... to n terms. When n is even, ^=(l-2) + (3-4)....to|groups = -1— 1— ....to - terms = - o > *he sum when w is even. SQUARES AND CUBES OF THE NATURAL NUMBERS, 105 When n is odd, Then and ^=l-(2-3)-(3-4)-....to n Y- • groups. = 1 + 1 + 1+ tol +^^— terms w + 1 = — 7) — > t'he sum when n is odd. 1/ , X 1 1. , 2w + l 1 = 4{l+(-irH2n+l)}, whether w be even or odd. um whether \ PILES OF SHOT AND SHELL. 127. Tojind the number of shot arranged in a complete pyramid ^ on a square base. Let n be the number of shot on a side of the lowest layer; then w- 1, n - 2, etc., will be the numbers on a side of the suc- cessive higher layers. The number of shot composing the layers will be n\ (n- 1)% etc., ending with a single shot at the top of the pile. Then aS= P+ 22 + . . . . (n - 1)2 + w» I _w(n+l)(2n+l) // 6 Art. 122 128. Tojind tJie numher of shot arranged in a complete pyramid whose hose is an equilateral triangle. Let n be the number of shot on a side of the base. Countinc 8 ^ \ 106 HIGHER ALGEBRA, the shot in the lowest, or < layer by rows we see that it con- tains n + (n-l) + (n-2).... + l=!!(!^±l). Similarly the {n - \f^ layer contains (!iziX^) -, etc. We have thus to find the sum of n terms of the series whose Ti*" term is ^{n^^n). Therefore 6 Art. 122, Ex. The number of shot in the successive layers, beginning at the top, are 1, 3, G, 10, 15, etc, which are called triangular num- bers for the same reason that 1, 4, 9, 16, etc., are called square numbers. * Jll ^^®' ^^-^"^ ^-'^ ^^^^^^ ^f'^ot contained in a complete pile ^ ij^upon a rectangular base. Let m and n be the number of shot in a side and an end of base. i-iAf^M:^^^-*^ There will be n layers, the top one consisting of a single row containing m-n^\ shot. Each succeeding layer will consist of one more row, and each row will contain one more shot than the preceding. The numbers in the successive layers will be the terms of the following series, .-. 'S'=(m-n-M) + 2(m-n-F2) + 3(m-r. + 3).,..M(m-n-f.ri) = (w-7i)(l+2 + 3-f-....ti) + (P + 22+32-f-....w2) _ (m-n)n(M+l) w(n+iV2w+l) 2 + 1 _ n{n-\ l)(3m - n+ 1) 6 "• If m = n the rectangle becomes a square, and this result re- viuces «> that obUiued for the number on a square base. that it con- tc. series whose rt. 122, Ex. ining at the ^•gular num- illed square ompleie pile I an end of single row 1 consist of 3t than the v^ill be the (m-n + n) V?) result re- SQUARES AND CUBES OF THE NATURAL NUMBERS. 107 130. To find the number of shot in an incomplete pile. Find I7 the preceding Arts, the number the pile would con- tain if complete, and from the result subtract the number re- quired to complete it. ^K.— Find the number of shot in an incomplete pile of six layers, there being 20 shot in a side and 12 in the end o£ the base. If complete the pile would contain -Ali^li^ ^ i274 shot. 6 6.7.37 // The number required to complete it is . = 269 shot. 6 Therefore the number of shot in the pile is 1274 - 259 = 1015. 3. '^^ f/ — ^^^' ^^^®" *^® number of shot in a complete square, or in a -m complete triangular, pile, to find the number in one side of the "^^ base. Let iVbe the number of shot in the pile and n the number in a side of the base. Then (1) In the triangular pile w(m + l)(n + 2) = CiT. Now n{n+\){n + 2) > n' but < (n+lf, therefore n is the integral part of the cube root of GiT. (2) In the square pile n{n+l)(n+Vj .-,3^^, therefore, as be- fore, n is the integral part of the cube root of SIf. BXBROISB XI, 1. Find the number of shot in a complete pile on a square base containing 20 shot on a side. 2. Find the number of shot in a complete pile on a triangular base containing 30 shot on a side. 3. Find the number of shot in a complete pile on a rectangular base containing 25 and 20 shot oir the side and the end respec 4 tively. ^ y 108 HIGHER ALGEBRA. 4. . low many shot in an incomplete pile of twelve courses on a triangular base containing 37 shot on a side t ^^^ 5. How many shot in an incomplete pile of eleven courses on a rectangular base containing 27 and 23 shot on the side and the end respectively ? 6. An incomplete square pile contains 225 shot in the top layer and 729 in the bottom; how many shot in the pile? 7. The base of an incomplete rectangular pile contains 800 shot and the top 450; the length of the base is greater than the breadth by 7 shot; how many shot in the whole pile 1 8. A triangular and a square pile of shot have each the same number on a side of the base, but the former contains ouly four- sevenths as many shot as the latter; find the number in each pile- 9. How many shot in an incomplete triangular pile of eleven courses, there being 166 more sliot in the bottom layer than in the top % 10. Show that the number of shot in a square pile is one-fourth the number in a triangular pile of double the number of courses. 11. If from a complete square pile of shot a triangular pile of the same number of courses bo formed, show that the remainder will just form another triangular pile. 12. The number of shot in an incomplete square pile is equal to six times the number required to complete it; and the number of completed courses is equal to the number of courses required to complete the pile; find the number of shot in the incomplete pile. 13. Find the number of shot in the rectangular pile in which the number in the lowest course is 600, and in the top ridge, 11. 14. How many courses must be taken fro-n the top of a com- plete square pile of shot to make up 2,870? How many more courses will make 12,040? 16. The value of a triangular pile of 16-lb. shot is $244.80; if SQUARES AND CUBES OF THE NATURAL NUMBERS. 109 ve courses on J.25 per cwt., find the number of shot the value of the iron be in the lowest layer. 16. A triangular and a square pile of shot have each the same number in a side of the base, and also one base row in common; the number of shot in both piles is 64,D75; how many in a row in the base, both piles being complete 1 17. How many 8-inch shells would there be in a square pile erected on the grouad formerly cover<^d by a square pile contain- ing 4,900 13-inch shells, both piles being complete? ^18. The numbers of shot in a triangular, a square and a rec- tangular pile are in A. P., the number of courses, w, being the same in each; show that w- 1 is a multiple of 3. If the rec- tangular pile contains 5,566 shot, ho-.v many does each of the others contain ? 19. Four equal square pi'es of shot are so placed that their bases form a larger square, and each pile has two base rows in common with the adjacent piles. There are 21 shot in the base roV of the large square; how many shot in the whole ? If there are 3,225 shot in all, how many in a base row of the large square? EXERCISE XII. MISCELLANEOUS EXAMPLES IN PROGRESSION. Sum to n terms the series: 1. 2^ + 4:' + 6K^. 3. a' + {a + lf + (a + 2y^'. 5. P.2-1. 22.3 + 32.4. .k"^ 7. P-22-,-32-.... C2. V+3'' + 6\/.. ^4. P-l-33 + 5'..'.'^ 6. w + 2(w-l) + 3(w-2).... 8. P_32-h52-.... n 9. Find the sum of the squares of n terms of the series whose ^ term is 2 - 3w. "^ 10, Tf S! ^o»^+«c +1 v\.-i-3 tut^ b1 ui« of tho first H natural numbers, find thosumsof,«?. + ^, + ....^„and-bf,y^-t-^,^.f.....^„^./ iH 110 HIGHER ALGEBRA. 11. Sum!^l±l%iiL?.\»*' + 33 • w + 1 n + 2 n + 3 ^^ terrain. 12. Give„that.+ 2{._^U3(. -1 V...^„^^, is zero, find a-. ^ ^^ '■ ** " ^^ 13. Given x, y, ^ in G. P.. y, z, 4 in H. P., ^, «•, y in A. P., find X, y and z, ^ 1 4. Sum to infinity the series : P_2' 3=_4' 5* 23 '2^ • ^-/ 5 52.5, 15 Divide the number 2--1 into n parts in the ratio of 1, J, 4, 8, etc. 16 Find the sum of the products of tl»e first n natural num- bers taken two and two together. 17. Show that —^ + __ is equal to 4, greater than 7 or 10, according as a, c, i are in A., G. or H. P. 18. The three sides of a triangle and the perpendicular from the opposite angle on the greatest side are in G. P.; show that the triangle is right-angled. 19. If 1, X, ^ and 1, y\ f be each in H. P., show that - ,/ ,/ ^, .are in A. P., and that their sum is ^ + y3, supposing . + ; not to be zero and ar and y not to be unity. 20. Sum 1.1.3 + 2.3. 5 + 3.5. 7 + 4. 7.9... .tonterms. 21. Let the sums of the squares of the roots of the equations, 3{^ + (m+l).} = l, ?{.r^+(m + 2).} = l, ?K + (.^ + 3H = l, be ^,^ and C; find the value of m so that A, B and G may be in G. P. ^ 22. Between the successive terms of an A. P. arithmetical means are inserted, one between ^,he first and second, two be- SQUARES AND CUBES OP THE NATURAL NUMBERS. Ill . to n terms tween the second and third, three between the third and fourth, and so on; find the sura of all the means so inserted, the first series consisting of w 4 1 terms. 23. SumP-2' + 3»-4» + ....to7iterms. 24. If ^1 be the arithmetic mean between a and i, A^ the arithmetic mean between Ay and i, and so on, show that A1 + A2 + ^„ = a + (w-l)6--— -. 25. Sum 2 .5 - 3 . 9 + 4 . IS - 5 . 17 + to 2n terms and to n terms. 26. If a, tti, ttjj «3' • • «n be in A. P., and a, i^, b^., .h„he in G. P., and if »•, the common ratio of the latter series, be equal to the common difierence of the former series, then {a,ar - ab,) + {aj,,r - a^b^) +,. . . (aX_,r - a^_J>,,) = ^'^ ~ ) . r-1 27. Show that the sum to n terms of the series whose (m-iy^ 1 term is m{m - 1) is equal to - of the product of the n'** terms of the three series whose (p - l)*"* terms are p-l, p and p + 1. 28. An A. P., a G. P. and an H. P. have each the same first and last terms and the same number of terms, n; and their r*** terms are a„ i„ c^; prove a,+i : fir+i = 6„_^:c„_,, and if ^, ^, C be the continued products of the n terms, then AG = B'. 29. Between two quantities an harmonic mean is inserted, and between each pair a geometric mean is inserted; the three means are in A. P.; prove that the ratio of the two quantities is 7±4 VZ: 1. 30. If a, 6, c, d are in G. P., then abcd\- + - + - + -A =={a-^.b + c + d)\ and a/zy* -I- M J- a/ hi. 112 HIGHER ALGEBRA. i| 31. If a, b. fw ; W^'^ I" ^' ^" ^""^ ^' ^ *^« arithmetic means be- Ween «, I and A. c. then A will be the harmonic mean between p 32. If a, 5, c are in A, P., .,, ^3, y in H. P., and - + ?:»« + ^ then ««, ij3, cy are in G. P. y « c o ' th!v' M^rT •''''S^'' *'" ^^ ^- ^-^ ^"^^ ^^ ^«^^ b« increased by 15 they will be, nH. P. The suu. of the numbers is 49; find tlfem infas !T f '* "^"""^".* ';^'*"^' ^^^^ ^^ "«^^*-^ --rd- mg as ai, a^, ag, a^ are m A P., G. P. or H. P. «x, Ai and a„ A, the arithmetic and harmonic means between m, a and g, n respectively, then aj^ = ^2 = a^, . ^ G.p';tL:'''^'^^'^/-^''«'^'^-«-^-'-<^««.¥.^in «:A:c=i:l ;! J ^ a Jl " " !r *'"'u"'^' °' *''° arithmetic means between two n«mbe.-s, and m the first of two harmonic m3ans between tie same two numfers, prove that the value of ,. doe, not To t tween the values of n and 9m. ftp then the ratio of the harmonic means between ; Td „ ;:nti:i:r"^' *° *"' """• -' *^ ^"'^*™ --- »* *-« -- 39 If J„ ^j . . . ^„ be the « arithmetic meajis, and H, . H, terms of the series whose r*"* term is Jld^::;;htv"''"* ^'' «-""*'« ■»-- *«'-» ''n-2--K-^(V-A„^)*}2. sou ARES AND CUBES OF THE NATURAL NUMBERS. 113 ive accord- 41. If a, i, c are in A. P., G. P. or H. P., then a** + c'»>24". 42. If ttj, rtj • • • • "n are in H. P., then a. are also in H. P. 43. If the squared differences of f?, 5-, r be in A. P., then the differences in order are in H. P. 44. If 1\ Q, li he the ju^ g-* and r"» terms of an H. P., then I I' -r' r^-jr^ j? Q' R' i~ \ F' ^ q^ ~^]\ F' ^"y 45. If n arithmetical and n harmonical means be inserted be- tween two quantities, a and h, and a series of n terms formed by dividing each arithmetical by the corresponding harmonical mean, the sum of the series will be { n{\ + •]■ 6(w+l)a6 46. Find the sum of n terms of the series, 1 + 22+3 + 42 + 5 + 62.... (1) when n is even, (2) when n is odd; find the w"' term. 47. Find the n^^ term and the sum of n terms of the series, 2 + 2(22 + 2) + 6 + 2(42 + 4) + 10 + 2(62 + 6) + 14 + 2(82 + 8).... (1) when n is even, (2) when n is odd, (3) when n is any positive integer. 48. Between each of the pairs of quantities, (a?, y), (ar, 2y\ (r, 4y), etc., are inserted m geometric means, and M^ is the w*" mean of the r*" pair; show that ~^^ = 2'»+i for all values of r Mf 49. There are n piles of stones placed in a straight line, the intervals between them being 3, 5, 7 .. .. 27i-l yards, and the piles containing 1, 2, 3 n stones respectively. How far must a nerson wnllr frk fipo+Ur.». +l»^,« „J i..:„4. 1 1 1 1 the end of the row at which is placed the single stone ] CHAPTER IX. IH SCALES OF NOTATION. 132. The basis of number is the unit or elementary number one; all other numbers are repetitions of this unit. The groups of units thus formed by successively adding one are known by distinctive names, and are represented by symbols. Thus " two » IS the name given to "one and one"; "three" is the name given to " two and one," and so on. These groups are represented by the symbols 1, 2. 3, 4 5, 6, 7, 8, 9, called digits, each of which represents a unit n.or(^ than the preceding. We have no symbol to represent «9 and 1"; we therefore give it a distinct name (ten), and represent it as a unit of a second order. To distinguish it from the simple unit we write the symbol beside it on the ric^ht When other units are added they take the place of the until two units of the second order are reached. Ten units of tho second order are expressed as a unit of the third order (named hundred), and so on, the order of any unit being determined by the position of the digit representing it, counting from the right. This method of representing numbers is called the Common Denary or Decimal Scale of Notation, and ten is said to be the radix, or base, of the system. 133. In like manner numbers may be represented by assuming a different radix and using the figures 1, 2, 3, etc, in the same sense as before. The number of independent significant symbols must evidently be one less than the radix. Thus if eight be taken for the radix, the figures 8 and 9 will bo unnecessary; but if twelve be taken, two new symbols must be introduced to repre- sent " ten " and " eleven " respectively. The letters t and e are generally used for this purpose. JL-i. SCALES OP Notation. 116 Tho exact meaning of each symbol employed should l)e care- fully observed. Thus in the common scale 365 means 3 times 10' + 6 times 10 + 5, but in the scale with radix 8 365 means 3 times 8' + 6 times 8 + 6; and generally, if Oo, a^, a^,... n^ denote the digits in order, be- ginning with the units, r the radix, then the number is «„r" + a„_ir»-^ + a^_,r^-^ + . . . . fl^r- + a^ + Oj where a„, rt„_i . .. .a^ are all positive integers, each less than r, but any one after the first may be zero. The radix itself is always represented by 10. 134. Our language, being adapted to the decimal mutation, is inapplicable to any other. Thus 26 in the scale of ei(,ht must not be read twenty-six, for it is not tioo tens and six, but two eights and six, or twenty-two. Since w.i have no words to desig- nate the numbers in the form in which they appear in the various scales, we read such numbers by naming the digits in order, giving the radix of the scale. 135. The various arithmetical operations can be performed in any scale on the principles which are employed in the common scale. But inasmuch as we are not familiar with the addition and multiplication tables in any but the common scale, we shall bo compellel tD determine the carrying figures by an indirect p* v/cess, as shown in the following example : Ex. i.— Multiply 2763 by 25 in the n«nary s«al«. 2763 \\ 15246 5636 72616 ^v 16 HIGHER ALGEBRA. In this scale we carry one for every nine. Multiplying by 5 we have r j s> j 3x5 = fifteen = 9 + 6 = 16. We therefore set down 6 and carry 1 ; then 6x5 + 1= thirty-one = 3 times 9 + 4 = 34, and so on till the multiplication is complete. The same method is followed in the addition. Bx. ^.-Divide 59^6^3 by 7 in the duodenary scale. 7[59<46e3 9e9285 rem. 4 The first two figures, 59, are not fifty-nine, but 5 times twelve + 9 = sixty-nme. Dividing by 7 we get 9 for quotient and re- mainder. Next, 6 times twelve + ^ = eighty-t wo; dividing by 7 we get eleven (or e) for quotient and 5 remainder, and so on. I ik?^^' ^"^ ^^■^'*^^* "* ^*''^'' number in a scale with a given radix. T^t K denote the number, r the radiv ^1 at T«t K denote the number, r the radix. r\ F Divide iT by r, the quotient by r, the second ^fe Q. Q, rem. p^ " Ih it Pt quotient by r, and so on until the last quo- ,. tient is less than r. Denote the successive ^ quotients by Q,,Q._.... ^„_,, ^,^^^ ^nd the re- mainders by ;,.„ 2h. ]h .... Pn.u as indicated r\ Q \ ' '««* '« in the margin. ^~- ^"-^ Then from the nature of division we have ^"^^QiV + p,, Q^ = Q,^r + p„ Q,= Q.^r+iK„ etc.. Therefore X^ q^^ ^^^^ = Qii^ +Pir + jt?o =Piif +Pn-\>'''~^ +;>ar-+jt?jr + »„,, which is the required number. When figures are used the radix SCALES OF NOTATION. 117 and signs of addi*"ion are omitted, and the last quotient followed by the several remainders are written consecutively. Ex. 1. — Express 3824 in the scale whose radix is 7. Dividing continually by 7, the quotients and remainders are as follows : 7 1 3824 7 1546 rem. 2 7]T8 " 7 [IT "1 1 " 4 The required number is 14102. Ex. 2. — Change 31247 from scale of eight to the common scale. First Solution. 31247 8 Second Solution. <[31247 < I 2420, <[14, 1, 7 6 9 2 25 8 202 8 1620 8 12967 Result in each case, 12967. The reason for the work in the first solution will be perceived by writing the number with the various powers of the radix thus : 3(8*) + 83 + 2(82) + 4(8) + 7. Now, 3(8*) + 8=5 = (3 X 8 + 1)83 = 25(8'), 25(83) + 2(82) = (25x8 + 2)82 = 202(82), etc. The division in the second solution is performed in the scale of eight; the reasoning is the same as that of the preceding example. . li 118 HIGHER ALGEBRA. Also, Ex. 1 may be solved by multiplication, like the first solu- tion of Ex. 2. The examples which follow should be solved by both methods. ^ E3XBRCISB XIII. tl. Add the following numbers, which are expressed in the nonary scale: 32078, 4135, 2057, 38725. r 2. From 20100431 take 14034324 in the quinary scale. ^3. Multiply 372^563 by 38 in the undenary scale. C 4. Divide 42765236 by 7 in the octenary scale. ^. Divide 32^^094 by 1 1 in the duodenary scale. C6. Divide 30102112 by 1323 in the quaternary scale. C 7 From 2061203 take 1626156 in the septenary scale, and multiply the difference by 506. 8. Find the G. C. M. of 323345 and 502341 in the senary ilBl, scat ^'^^ *^' ^' ^" ^^* ""^ ^^' ^^' ^^' ^^' ^^' ^^ ^^ *^« "^^^^^^y ^ 10. Express a million in the nonary scale. scale' ^^^"^' ^^^^^^' ^''"' *^'' '''■^' °^ '^"^"^ *^ *^" ^«°^°^°" X>2. Change 8032765 from the nonary to the septenary scale. ' QP. Multiply 4541 octenary by 21301 quaternary, extract the square root of the product, and give the result in the septenary^^- ^ 14. Extract the square root of 11000000100001 in the binary ,15. Show that the numbers 121, 144, 1234321 are perfect s-?Juares in any scale whose radix is greater than 4. \ ,, ^^; ?i*!" TT]^^^ ]! 2' ^' «' ^*^-' lbs., which must be taken to^ wuign o'Ji iDs.i io'lo lbs.? n SCALES OF NOTATION. 119 common 17. Of the weights 1, 3, 9, 27, etc., lbs., which must be taken ^ to weigh 1852 lbs., one of each kind only to be taken, but to be ^^^y placed in whichever scale is necessary 1 ^'■ (' 18. In what scale is 182 represented by 222 1 ^ ,-»^19. Find the scale in which 519 is the square of 23. v20. Find the scale in which the product of 32 and 25 is 1163 21. Show that 1367631 is a perfect cube in every scale whose /H>-wv/ radix is greater than^ 7. 22. Find the scale in which 12736 is represented by 30700. 23. In what scale is 511197 denoted ky 17463351 d / lyii. Add the following fractions in the scale of eight: 3 ^ 13 _7^ 5 4' 10' 20' 14' 30* 25. A rectangle is 13 ft. 6,^ in. long and 10 ft. 4 in. Mvide;6'y:^ find its area by multiplication in the scale of twelve. 26. Find by division in the scale of twelve the height of a c^>vvi right-solid containing 282 cu. ft. 705 cu. in., whose base contains ^ 24 sq. ft. 5 sq. in. __ 'Ia/ J / ■Koti^^looSt. 137. Radix Fractions are a series of fractions whose de- nominators are successive powers of the radix of the scale, and whose numerators are each less than the radix. Radix fractions in any scale correspond to decimals in the common scale, and are distinguished from integers in the same way, by being preceded by a point, which may be called the radix point. Thus, in the octenary scale, 237-6574 = 2(82)^3(8)^7#-*-AlA-i. ! 120 HIGHER ALGEBRA. but since in this scale seveii is the highest number expressed by one symbol, the above must be written 237-6574 = 2(10)2 + 3(10) + 7 + —+ — + —+-— since the radix is always expressed by 10. -^ t'-'138. To transform a given fraction into radix fractions in any /Y proposed scale. n ^^^^^ fraction and r the radix of the proposed scale. Multiply m by r and divide the product by n', let q^ be the quotient and r^ the remainder. Multiply r^ by r and divide by n; let q., be the quotient and r.^ the remainder, and so on. The quotients - tained by subtracting in order the digits already found from the Ny radix, minus one. With the notation of Art. 138 suppose a remainder n-m occurs. (»i - 111) r Then nr — n<7i n ■ = T-\ n — r. 1\ n 'n which shows that the next quotient is obtained by subtracting the first one, q^, from r-\, and the remainder, n-r^, being of the same form as n - m, the next quotient must also be found by subtracting q^ from r - 1, and so on. We thus double the number of digits alrettdy found. In the Ex. of Art. 139, if the process be continued two places further the remainder 8 ( = 11 in second solution) occurs, and it will be found that the succeeding figures are obtained by sub- tracting those already found from 6. The complete result is 5 Yg = -245631421035, both sides of the equality being expressed in the scale of 7. In this example it may be observed that the number of figures re- peating is one less than the number of units in the denominator, but in the common scale the number repeating is onh/half as ereat. y^ / .J 141. The difference between anij number and the sum of its ligits is divisible by r-1 where r is tlie radix of the scale in which tlie number is expressed. Let JV denote the number, a, l,c.,.. the digits in order be- ginning with the units, S the sum of the digits. Then J^=a + br + cr'^ + d'r^ + , , ._ , S=a-^b + c + d+ SCALES OF NOTATION. J23 ==(''-'^W' + c(r+l) + d(r^ + r+l) + ,,,,^ Therefore »•- 1 is a factor oiJV-S. ^Ae same dibits is divisible hjr-l. ^ For let iT, and i\-, denote the numbers, S the sum of the dibits ' in either case : then since N ^ ^^a \r o \ ^"^ ^ ' ' ^*' ~ *)°'^-*i--'l^i IS divisible by r-l. Let iT denote the number a h ^ +v j- -^ • Then ^=a + br + cr'' + dr' .. . . ^ = a-b + c-d+.... = a multiple of r+1. Therefore if i> is a multinle of ^ + 1 so al.o ,'. - ri,- 111 ^'11 \ HIGHER ALGEBRA. 143. If the sums of the digits of two numbers in the common icale be separdtely ilivided by 9> and if the product of the two remainders again be divided by 9, the last remainder will be the same as that obtained by dividing by 9 the sum of the digits of the product of the original numbers. Let iVi and K^ denote the numbers, rj, r^ the remainders ob- tained by dividing the sums of their digits by 9; then ri, r^ are also the remainders when iVi, N^ are divided by 9 (Art. 141, Cor. 1). Let g'l, q^ be the quotients. Then -^i = %H-ri, and N.,= ^q^ + r^; ^ = a multiple of 9 + r^r^. •herefore the remainder, when 7-1^2 h divided hy 9, is the same as when the product iVjiVa is divided by 9; and this is the same as that obtained by dividing tiie sum of the digits by 9. The above is called the ''liule for casting out the nines." It will be observed that the rule fails to detect any error which does not affect the sum of the digits in the result, or which changes their sum by a multiple of 9. EXERCISE XIV. 15 Ojt. Transform — from the common scale to radix fractions in the scales of four, six, eight and twelve. 11 5 3 ^^ 2. Transform — , --, — from the scale of six into ordinary decimals. f^ 3. Transform -15625 and -2083 from the common scale to scale six. nmls. fractions in V S«ALfiS 4P NtTATl^N. jAg .J,!""'"""" '''''■''' ''"'" *"*> -'« <" ■"»*.,«,« »c.le of 6. What fractions are equivalent to -224 anrf -lilic' • *t scales of twelve and eight respectively J " '" *'"' otli if"!!"',™'" °' :f "" =™'V--; of -15 in scale ei^ht. of 91 in scale eleven; and of -(9724 in scale twelve of se'J""'"™ "'""■"' '™"" *•■« -'e of eleven to the scale ». Transform «.. and le-lee from scale twelve to scale eleven. ^riL'rr'"' '°^"" "''" S ^ --% oxprised by finite r«,i. sent!d b;;'^;"'^ '^ '°'"'''' °' ^™"' '- oorrectjyj • 13. Transform •111101011 fr/^T« +1. v ' ^ scale four ? ""^"^ ^^^^*^ ^"^"^^ b« required in A aivisiLity o^:ieX7 itr "^ " '"""' " "" ^°' *"« -l^"crd t;:;i:^f""'^ °' "-»•=- ^^ -« -nary vf d lylddi't'Vr'b'^V;-^*?""'' ""'' ''^ '"S"^ ■'- - -Plenary stle "^ ""'"« ".'""' t^-'^orming into the to ":J!:Z-TT '"'''7 '^.f- "' ^lenominator be reduced X ceed;ither;(ri)r ;;"^r' '""" ""* ^"""^ '^''"°' ''-^ 1^6 hlOHEft ALGEBRA. 18. Prove that in every scale whose radix is greater than 8 the number represented by 1 1088 is divisible by that represented by 12; that the first figure of the quotient is one less than the radix; and that the last two digits are in every case the same. ^19. Prove that any number expressed by four digits in the common scale is divisible by 7 providing the first and last digits are equal, and the hundreds digit is twice the tens di«it "^0. If a number in the common scale is divisible by 3, the numbers expressed by the same digits in the scales of four, sLven and thirteen are also divisible by 3. 21. In a scale whose radix is odd, any number and the sura of its digits are both odd or both even. 22. If S, be the sum of the digits of a number, ^, expressed m the septenary scale^ and 2S^ the sum of the digits of 2JV ex- pressed in the same scale, then the difference between S, and S is a multiple of 3. ^ 23. Prove that the squ.^re of rrrr in the scale of s is rrrqOOOl, where q, r, s are any three consecutive integers. "^ 24. In the scale of notation whose radix is r, ahow that the number (r^- 1)(^- l), ^hen divided by r - 1, will give a quo- tient with the same digits in the reverse order. (I 25. If from any number expressed in the nonary scale is sub- tracted the sum of every third digit beginning with the units twice the sum of every third digit beginning with the tens, and four times the sum of the remaining digits, the remainder i. divisible by 7. ^6. Show that any number of six digits in the common scale whose first and fourth, second and fifth, third and sixth digitl are alike, is divisible by 7, 11 and 13. 27. A certain fraction is correctly represented by -21 in the scale of X, by -27 in the scale of y, and by -5 in the scale of x+y express the frn.of.inn as an ot./i;»,a-,. j„— •„- i x ^'^ SCALES OP NOTATION. 127 sClhe digits of a number are added, the digits of this sum are add.d, and so on until the last sum is a single digit. If this operation be performed upon several numbers, and then the same operation upon the resulting single digits, the final result will bo the same as that obtained by performing the same operation upon the sum of the original numbers. ^^- ^^ (TTiyj ^ reduced to radix fractions in the s ale of r, show that the period which repeats is composed of .ero followed by the digits in order up to r - 1, omitting the digit r - 2. .,30. If jy^ p,,p,..., be the digits of any number beginning Vith the units, prova that the number is divisible (I) in the common scale by 8 iip, + 2p, + ip^ is divisible by 8: (2) in the scale of twelve by 8 if 4p, +p, is divisible by 8; and by 2, 3 or 6 providing p, ,8 so divisible. Give similar tests for the divisibility of numbers in the scale of twelve by sixteen, eighteen, twenty four and seventy-two. ' j 31. If the digits of any number in the common scale be divided mto groups of six digits each beginning with the units, and if the digits in order of each group be multiplied by 1, 3, 2 6 4 5 re- spectively, and the sum of the products bo subtracted from the given number, the remainder will be divisible by 7. Give a/ similar theorem when 13 is substituted for 7. CHAPTEK X. SQUARE AND CUBE ROOTS, AND SURDS. SQUARE ROOT OF NUMBERS. 144. Practical rules for the extraction of the square and cube roots of numbers are given in all works on arithmetic, but the reasoning employed, being algebraical, is not suitable for students at that stage of their studies. We shall now, therefore, deduce the reason for the ordinary rules for extracting the square and cube roots from the methods given (Part I., Chapter XI.) for the extraction of the corresponding roots of algebraical expressions. Examples of whole numbers only have been given, but the princi- ples are equally applicable to decimals. 145. The integral part of the square root of a number less than 100 consists of otic digit; of a number between 100 and 10,000 consists of two digits; of a number between 10,000 and 1,000,000 consists of three digits, and so on. In other words, the square root of a number consists of one, two, three, etc., digits, according as the number consists of one or two digits, three or four digits, five or six digits, etc. If, then, we divide the digits of the given number into groups of two digits each be- ginning with the units, the number of groups will give the num- ber of digits in the root. This gives us the highest power of 10 contained in the root, which, multiplied by the largest integer whose square is not greater than the left-hand group, gives a first approximation or is the first figure in the required root. SQDAIlfi ASh CU6I1 ROOTS, AKD SUHCS. 12& 146. Let JV denote a number whose square root is to be found, and let a denote a first approxinmtion found by the last Art.] and let x denote the remaining part of the root. Then JV'= a» + 2aar + ar' or JV- a" = 2ax + x\ from which equations the value of x must be found. Now, neglecting x\ which is considerably less than lax, we get •* = ~2a~* ^* ^» ^ *^® ^'*^* figure of the quotient, followed hy the proper number of ciphers; add it to '^a, multiply the i-esult by a?! and subtract from N-a^, and we get N-a^- 2axi - ar,« or N-{a + x^f. Denote a + x, by ai and the remaining part of the root by x^, and proceed as before. In practical work it will frequently be found that 2axi + xi' is greater than JV - a''; in such cases a smaller integer than x^ must be taken. Ex. — Find the square root of 119025. Dividing the digits into groups of two digits each we see that the root ust contain three digits; and since the greatest integer whose square is less than 11 (the left-hand group) is 3, therefore 300 is a first approximation. Then 11 9025 = (300 + xf = 90000 + 600ar + x\ :. 600.r + ar2= 29025 or ar = 40; then (600 + 40) x 40 = 25600; 29025 - 25600 = 3425. Again, 680;ir + ar» = 3425 or ar=5, *^^«" (680 + 5) X 5 = 3426, which shows that 345 is the root required. . , >'*«:.*^ ISO HlGfiER ALGEBRA. The preceding work may be arranged thus: 300 300 X 2 = 600 40 640 340 X 2 = 680 5 685 11 90 25(300 + 40 + 5 9 00 00 2 90^5 2 56 00 . ~3r25 34 25 The student should compare the above with what precedes then omit the zeros and arrange the work in the usual way al below, and observe that the final operation is only a convenient arrangement of the process first given. 3. 64 119025(345 . 9 . 290 256 685 3425 3425 147. When n + 1 figures of the square root of a number have been found by the ordinary process, n more may be found by dividing the last remainder by twice the root already found, tlie whole root consisting of 2n + l figures. Let a denote the part of the root already found, x the part to be found, K the given number. Then N^a-' + 2ax + x''- therefore N-a'^ = 2ax + x^ and 2a Now, iT- a^ is the last remainder, and this divided by 2a, t.e., twice the mot already found, gives .r the part to be found increased by ~, which we shall show to be less than unity. SQtJARE AND CUfiE ROOTS, AND SURDS. 131 Now, a ' ontains n + 1 digits followed by n digits, and X contains n digits, /. z< 10". a>102"; Therefore a;^ 2a *^ 2 (10)^" 1 which proves the proposition. If the number is not a complete square the above demonstra- tion fails. But the quotient obtained by this method in all cases differs from the true root by less than a unit in the last digit; it may therefore always be used for finding approximate values of the roots of surds. Similar remarks apply to the theorems of Arts. 151 and 152. CUBE ROOT OF NUMBERS. 148. The integral part of the cube root of a number less than 1,000 consists of one digit; of a number between 1,000 and 1,000,000, of two digits, etc. Therefore the cube root of a num- ber consisting of one, two or three digits, consists of one digit- of a number consisting of four, five or six digits, consists of two' digits, etc. If, then, we divide the digits of a given number into groups of three digits each beginning with the units, the number of groups will give the number of units in the root. This gives us the highest power of 10 contained in the root, which, multiplied by the largest integer whose cube is not greater than the left-hand group, gives a first approximation to the required root. 11 149. Let JV denote a number whose cube root is to be found, and let a be the part found as above, and let x be the remaininc^ part of the root. ° Then Ji^a^ + Sa^x + Sax^ + x", = {:ia? + ^ux + x:^)x. or 132 MlOHfiU ALGEBttA. 'f ! ■I' I- ii NoglootmK tho tonus 3...'^ + ^' which are less th.ui .'Ja'^.r. wo got .r = --^^^ - . T^t .r, 1,0 tho first figure of tho quotient, followed by the proper number of ciphers; substitute it for x in tho ex- pression, (M.^'.r + 3..r + .r^).r, and subtract it from the In^t re- nuunder, viz iT-a^ giving ir-(a + ..)'; for « + .r, write .„ and proceed as Ix^foro until there is no remainder, or until the root h,is be«,n found to tho required degree of accuracy Should the numerical value of (3«^ + 3«^, + ^r)'". be gimter than JY- a, an integer smaller than u:, must bo taken. 150. In practical work the tedious part of the operation con- sist! in calculating tho values of tho trial divisors, 3a\ 3« ^ etc and of the co.nplote divisors, Sa'+3ax + x^, etc. By properlv arranging the work their values may be calculated in succession the value of each being used in finding the value of the followin.^ one. The method will be evident f,x,m the arrangement of the quantities in the two columns below. The first approximation to the root IS det.oted by «, and tho successive additions to it by f>, c, etc., which for distinctness may bo called quotients First Column. 3(1, 3a + h, 3« + 2A, 3(a^ and «>.r; x \2a + x) .r"(2a + a:) x^ N+ 2«3 3a= a' If « contains n digits and a,- contains r digits, *^en «>(10)»+'-i and ar<(10)'-. Therefore arJ -^<- (10) ,3/- (10)' «^'^(10)-<"+''-i''^(10)2("^i)* And the error is less than a unit if r is not greater tlian 2(n - 1). NoTE.-In practical work a" is found afc onco by subtracting ,ho last re- niainder from the given number. -fi'a;. — To find the cube root of 7. By the ordinary method we find the first tliree figures to be 1-91, and the remainder, -032129. Then and Therefore «3= 7 _. 032129 = 6-967871, -032129x1-91 7 + 2(6-967871) '"'^^^^^^' ^7"= 1-912931. SURDS. 153. The most important properties of surds have already been explained (Part I., Chapter XIIL); we now discuss a few more complicat/ + z==a, it would be erroneous to infer that the given expression has no square root. The correct inference is that it has not a square root of the assumed form; it may have a root of a_different form^ For example, consider •he expression, 12 + 8V'2 + 6\/3h-4a/6. Proceeding as before we obtain the equations, 2a/^ = 8v/2; 2a/^ = 6a/3, 2\/'^ = 4V6, which are satisfied by :r = 5^, y = 6, « = 4^. But these values do not satisfy x + i/ + z=l2, therefore the_square root of the given expressions is not of tjie form Vx+ Vy^-V^. The correct root IS 1 + \/2+ '/3+ \/6; but no direct process can be given for obtaining the root in such cases. It will be instructive for the student to write out the square of each of the expressions, x+Vy+ V7+ V^z, \^7+ \/y+ \/z~+ V^, m+ Vxy+ Vyz+ Vzx, and to observe that the result in each case is of the same form as that of the preceding Art. But if we attempt to obtain the root of a numerical example by using any one of these results, we shall find the resulting equations too difficult for solution. 10 ^is^'snmi 138 HIGHER ALGEBRA. lif 2/- 157. J/ ^a-^ V~h = x+ Vy, then will ^ a- V'l=.t-V For by cubing we obtain a+ \^ b = x^ + ^x^Vy+ 'dxy + yVy. Equating the rational, and also the irrational, parts, we Iiave a = x^ + ^xy, V b = Zx'^V y + yV^. Therefore a - V'b = x^ - 3x^ V y + Sxy -y^y, or '^a-Vlt^x- Vl/. Similarly it may be shown that if ^ «+ \/6 =a;+ A/y, then ^ where n is any positive integer. a- V h-x- Vy, 158. To extract t/te cube root of a binomial quadratio snrd. Since {x^ ^^f = x?^- 2>xy + (3.r- + y) V y, and {Vlc+V]/f = {x + 3y) ^/x + {y + 3x) V y, we see that the cube of a binomial quadratic surd is a quadratic surd of the same form. We therefore reduce the given surd to its simplest form, and assume its root to be a similar surd. 1. If one term be rational Assume Then therefore ^«+ 7iVb = x-\-ys/b. y rr ^ a - nVb^x — yVb^ \^a''-n'b = x^-by\ Cubing (1), and equating the rational parts, we get x{x'+Uy'') = a. and from (2), x"^ - by^ = c, (1) Art. 157 (2) (3) (4) where, (^ = a'^-n^b. SQUARE AND CUBE ROOTS, AND SURDS. 139 2. If both terms be surds: Then, as before, x(ax^ + Si?/^) = „i, where c-'' = m-a - n-h. (6) Now, If the ongznal surd is an exact cube, and its coefficients a e positive integers . and y must also be positive integers, and 3 equal to an integral quantity. If the coefficients are not in- tegral they may be made integral by multiplying through by the thiTLtor ' *^' '""*' """'^ ^' "^^"^^^"^ ^y *^" ^°°<^ «^ The values of ;. and y must be found from (3) and (4), or from (5) and (G), by trial ; but since they are positive integert, in mo t cases this may easily be done. The numerical examples which follow show the best method of proceeding. I^x. 1. — Find the cube root of 207 + 94 V'h. ^207 + 94 VT = ^.+yv/5: Assume Then from which or and ^207 - 9475 = a: -y ^5, .f2 _ 5y2 _ ^^'(207735(947, (1) (2) 4^2+15^2)^207 From {llx-^t^y^-. U. Giving y the values 1, 2, etc., in suc- cession we find 2,= 2, .. = 3_satisfies this equation and alL equa- tion (2); therefore 3 + 2 V' 5 is the root required. Bx. ^.— Find the cube root of 921.7 - 4122 VJ. The equations are : x"-5y'^-n, .r(.f=+15/) = 9217. (1) (2) 140 HIGHER ALGEBRA. The values y « 2, x = 3 satisfy (1) as before, but do not satisfy (2). Giving y the values 3, 4, etc., wo find y = 6 gives a com- plete square, 169, for the value of x-; therefore 13 - 6 a/ 5^ is the root required. Bx. ;?.— Extract the cube root of 430 \^~2+ 324 \/ "3. Assume ^430V'2+324\/¥ = a?A/¥ + yV'3'. Then ^430^/2 -324 a/"3 =a; V'2 -y v^S, therefore 2;r» - 3^^ = v" (430 v" ty^32iV3)- aud = 38, .r(2;c^+V) = 430. Giving y the values 1, 2, etc., until an integral value is also obtained^or r, we find y = 2, a; = 5 sa^'sfies both equations; there- fore 5 V^ 2 + 2 V 3 is the root required. 159. The student's progress in many parts of mathematics, especially in the solution of equations and in Trigonometry, will be much facilitated by a thorough knowledge of surds. We therefore give a large collection, chiefly selected from examples which have presented themselves in practical work. EXISROISE XV. 3 1. Find the square root of 10, - and 3-1415926536, each to ten decimal places. 2. Find the cube root of 2, -2 and 1-9098593172, each to ten significant figures. 3. Find the value of -'^10 + 2 a/ 5 and of ^_\ each ^ 2a/2 seven places of decimals. to SQUARE AND CUBE ROOTiS, AND SURDS. 141 4. Find the square root of 8 V' 2 + 2 a/30, 7v/'3-12 -1 V2" aiitl nVm~2mVn- f). Find the square root of 16-2V2(>-2v^28+^l40andof40+12v/6+8l/T0 + 6v/T5. 6. Find the square root of 21+3V'8-6i/3 -61/7-/24- 1/56 + 21/21. 7. Extract the cube root of 7 + 5V'2, 72-321/5 and 1351/3-871/6. 8. Simplify {1351-780v/3'}*-{26 + 15i/3}~^. 9. Divide V3 + 3 by 31/3 + 5 and .r-;r' + 2ar VTT^ by 1 +a-- 1/!^.. 10. Simplify (.r- 1 + Vl){x- 1 - 1/ 2)(ar + 2 + 1/ 3)(.r + 2 - 1/3). 11. Simplify li:^^ + (lz.^^:lKLt5J^ 12. Simplify {Vb+ V^3+ 1/2"+ 1)2 + (1/5+ V3_ y/g"- 1)2 + (v/5- 1/3+ 1/2- 1)2 + (1/5'- i/y- 1/2 + 1)2. T? -p ( ^3"+ 1/ 5)( V' 5"+ 1/ 2"-) tion with a rational denominator. V'S + I 14. Show that -;_-^__- = (^2+^3X/2-l). 15. Simplify 1- x^+f 142 HiaHER ALGEnRA. 16. Simplify ^ 2 ^ 2 V^ 2 X ^ 2 v' 2 i/ 2 -h ^ 2 ^^ iTf. 17. Show that ^'S+V5- ^5- V'b = {8-2 ^10 + 2 Vb) * 18. Find the continued product of the six factors, x""- ^ 3 + 1 ^ v" 3 - 1 ar+l, .^2 T^-x+\, 3-'' + .rV'2+l, V'i .r' - ar v/ 2 + 1 v/2 ar+1, a;2 + -^x+\. x'^-\ V2 ' a/ 2 19. Multiply ar2-(i?'2-l)a:+ v'I+^2+1 by ar+^2"-l. 20. Divide 2.c3-6ar + 5 by a'^^^- v'l+l. 2^2TvT 21. Simplify 22. Simplify ^3 - >/ , and ^/2+ ^7-3\/5 4- '^6-4-v/2 (48^ + ^v/T5)^ + (48^_^^15)i a/ 20 23. Show that ^«2 ^ ^^4^,2+ v'/^a ^ v'(7Z* = (a§ + i§)^ 24. Simplify ^16 -QV7 VZ + ^1 ^and2v'24v^l8-^?. ^2 l/l2. 25. Find the value of ar* - 3 f 2 x when ar = — it^ 26. Find the value of x'+Sqx when 27. Simplify (1 + \/ 2 - v/3) ^"^TTf - 2 J2 - -^ti ^ v" 2 SQUARE AND CUBE ROOTS. AND SlTRDP 28. Simplify 143 A'G-15v/3 (1/2 + ^3)(v/3 > ^6)(v^6 + v/2) "^'5^0: ^I^THTI * 29. Show that [ i ) =64|2ViO-2v'5- ^10 + 21/5}. 30. Extract the cube root of dah"^ + (h- + 2U^) Vf^^sl?. 2+V3 2-i/3 31. Show that :^ + ^^2+ ^2 + V'6 a/2 - ^2 .= v^2. vs 32. Show that ( a/ 3 + V2 -1)^2 + V2=2J2 \ ^^+^ IOa/2 33. Find the value of V2 a/18- ^"3+ a/5 a/8+ v'3- a/5 to five places of decimals. 34. Show that \2 a/2 + A/3 + 1/ \2v'2+ A/3-1/ difference of two simple surds. 36. Simplify > (^3 + a/2- 1)^2 + a/2 _, v^FiF- -^^^15625 2^4 + a/6 + a/2 37. Find the value of and ^270 + v'33-75 2P9 o7~7Tn (^ a; 4- A/ a;) when ar= ( 2{a- + b')^ ^ \a-bj sa CHAPTEK XI. IMAGINARY QUANTITIES. 160. From the meaning given to Multiplication the product of two equal factors has been shown to be essentially positive, and the square rr .t of an algebraical expression has been defined to be one of two equal factors whose product is the given expres- sion; from which it follows that to speak of the square root of a negative quantity is a contradiction of terms, and is therefore an absurdity. For this reason the terms, " impossible," " imaginary," "not real," have been applied to symbols denoting such contra- dictory operations. When, however, the proper meaning is at- tached to the symbol V -I, which may be taken as the repre- sentative of all the so-called imaginary expressions, it becomes quite as real and intelligible as any other symbol whatever. But the words "imaginary," etc., are too firmly fixed in the language of mathematics to be changed, and this is the necessary and sufli- cient reason for their being retained. It is customary in mathematical works to assume that ima^^in- ary quantities are subject to all the operations of elementary algebra without assigning any intelligible meaning to either the symbols of quantity or the operations performed upon them; and this course was adopted in the brief treatment given in Part I. "We shall now give a rigorous investigation of the truth of what was there assumed, according to the meaning which we shall assign to t!ie symbols of imaginary quantities and the operations to be performed upon and by them. 1C1. When a quantity is multiplied by a negative number dif- ferent from unity, two distinct operations are performed: (1) the IMAGINARY QUANTITIES. 145 magnitude is increased or diminished, and (2) its relation to some other quantity is changed, i.e., it is changed from positvve to nega- Uve or v^ce versa. For the present leaving out the numerical value and taking - 1 for our multiplier, let us carefully examine Its effect. By multiplying by - 1, a number denoting cash in hand or money due to me is transformed into a number denotin<. a debt due fiy me; a number denoting time reckoned after a given event into a number denoting time preceding that event; and a number denoting a distance measured in one direction into a number denoting an equal distance in the opposite direction Now the question arises, Is it intelligible to speak of performing a part of any one of these operations ? or, in other words, Is there any mtermediate stage between positive and negative ? In the case of distance and direction there is; in all other cases there is not- consequently i/ - 1 has an intelligible meaning when applied to space, but IS unintelligible in connection with any other kind of quantity. J^^"«^^* ^^^^ ^^ ^ '''''^^' "^"^^"^ "• ^"^^ ^^^ diameters AUG, BOD at right angles to each other; then if OA be denoted by +a, OG will be correctly repre- sented by -a; therefore OA multi- plied by -1 becomes OG. In the process of changing OA into OG, con- ceive that OA revolves around 0, through the semicircle ABG, and con- sequently passes through the position OB. Now, distance measured in direc- tion OB is neither positive nor nega- tive, it is the intermediate stage re- ferred to in the last Art. To turn OA through a right angle into the position OB is to perform half the operation of multi- plymg It by - 1; for if the operation be repeated upon OB the result is OG, which is the result obtained by multiplying OA by - i. Now, to multiply twice by the square root of a number gives the same result as to multiply once by the number; there- 1 mmsmmmmiMuemiAu ■^-~.»-^>:,;ri-i;nffl'^-°'": 146 Higher algebra. I fore, as a multiplier, the square root of a number bears the same relation to the number itself as the operation of turning a line through one right angle bears to the operation of turning it through iioo right angles, which is equivalent to multiplying it by - 1. For this reason it is convenient (and reasonable) to define V - 1 to be the symbol of the operation of turning a line f. Jm its original position through one right angle. 163. The operations symbolized by V~^ may be performed upon the result of a previous operation of the same kind, thus: a/^ . OA = OB, -v/TTI . OB = OC, V~ , OC = OD, V~^\. . OD = OA, etc. If, now, we denote one of these quantities, OA, by a, and the number of operations performed upon it by an exponent affixed to the operator, we shail have the following results: OB=^~I~\,OA=. ^^l.a-, OC = -/TT. OB=.^^—lY,a^ -a, since 0C= - OA; = - V ~ I . a, since OD = - OB; OD V -\.0C ={V -If a OA = V-\.OD^{V -\y.a^+a. Since (a/^)*. « = «, the symbol {V^^ in connection with any quantity may be introduced or omitted any number of times without producing any change whatever. This principle enables us to give_at once the result of any number of such operations Thus ( V - If. a = a, V-l.a, -a,ov - -/— . „, according as n, when divided by 4, gives 0. 1, 2 or 3 for remainder. 164. It should be observed that it would have been equally correct to assume that V^ - 1 as an operator turns a line in a direction opposite to that which we have chosen. Had this been done the symbols representing OD and OB would simply have IMAGINARY 8. 0. Siiico imaijitiary miinhors and n>a.l numluM-s donoto distances (voux a HximI point, along two Vmrn at right anghvs to each other, an iniaginary nundun-^can novcr bo oquivalont to a roal number. If, thoi-oforo, a + 6 1/ - 1 X. 0, thon n and b must separately vaniali. COMPLEX NUMBERS. 170. AVo hav»> now assigned an int(>lligil)hi meaning to imagin- ary quantities, and have shown that, with this nuviuing, two such quantities may bo addcnl or subtracted in the same way as real quantities. Wo have also assigned a nuvining to the operations ol -MiUiplieatioa and division of two quantities, providing any qu^' uty is wholly real or wholly imaginary. It remains to do- ternuno what nu\vning should bo attached to tho sum of a real and ai\ inuigiiuiry quantity, ami to tho operations of nuiltiplica- tiou and division with such combinations, in order that tho whole nuiy be in harmony with the definitions ami rules of Elementary Algt>b.-jv, and with what has already been dotormiued with regard to pure imaginaries. 171. A Complex Number is tho sum of a real and an imaginary nundnM'. Its general foiin is a + il,^ where a and b may have any numoric^il value, positive or negative, i.e., a and b n\ay l)0 any (piantities which do not involve tho imaginary sym- Ih>1 i. The exact meaning of tho word "sum" should bo noted, both when it refei-s to quantities (actual qmuitities, not to their repre- sentativ.) symln^ls) and to algebraic expressions. Tho sum of two quantities is the quantity formed by combining tho given quanti- ties. Tlie sum of two algebmical expressions is tho combination of syml)ols which convctly ,.op.nv.ont3 tho quantity formed by combining tho qu;uitities represented by the given expressions. if A I * ii X^ — X IMAGINAUY QUANTITIES. 151 172. To r^presrut the sum of a real and an hnaghu^ry number, t.c, a annplex number. ' Other. Ixit mil numbers ho meas- urod from (? in diroctious OX ov 0A'\ nccordiiig as thoy aro posi- tivooriiogativo; thou imaginary imm1)crs must be moasurod indi- rections or or or, according as the sign of the real factor is posi- tive or negative. From tiike CM in direction O^and a units in length; from M take MP, in direction OF and i units in o.^th; then ^1/ and 3fP, are correctly represented in magni- tude and direction by -|-. and +ib respectively. Now, the re- sult of a motion from to M, followed by (or plus) a motion from Ji to J „ IS the same as a motion from to P,; therefore with this extension of the meaning of the sign -f. OP, is the correct representative, both in magnitude and direction, of the complex number a + ih. ^ Similarly 0P,=. -a + ib, 0P,= -a-ib, OP, = a-ib. It IS evident that the point /', might be reached by first meas- unng i units in direction OV, and then a units in direction OX liierefore a + tb = ib + a. H ^^^;^vf''' ^r^^^ ^''^- «J^«"ld be carefully compared with tlie addition of positive and negative numbers (Part I., Art. 34) To add a positive units and b negative unit, we measure a units in the positive direction, and from the extremity of this line me^isure i units in the negative direction. The distance and drrectran of the extremity of the latter line is taken for the sum of the two numbers; and this is precisely the metliod adopted in the preceding Art. In both cases the sum nf f.I,« L.^rfjJ^f .i-_ two lines added is greater than tlie length of the line'^taken for their sum; but in both cases direction as well as length is con- ■'•'*• ''^■■'-"••"■•m'lfinri'nti riiH"f 152 HIGHER ALGEBRA. sidered in the addition, and it is this element which causes the difference. 174. The M odulu s of a complex number, a + ih, is the posi- tive value of Vd^^b^ which may be considered the absolute, or numerical, value of the expression. It will be observed that it represents the length of the line OP, without regard to direction • so thatifa circle be described from as centre, with radius equal to Vd'+b\ an indefinite number of complex numbers may be represented, each of which has the same modulus, viz., the radius of the circle. 175. The Argument of a complex number is the angle through which the line of positive, real units must be rotated to corre- spond with the line denoting the complex number; its magnitude 18 determined by the signs of a and h, together with their relative numerical value. The four numbers, a + *A -a + t'6 -a-ib a-ib represented by OP,, OP,, OP,, OP,, have each the same modulus, but different arguments. Sometimes it is convenient to consider the argument negative. Thus the argument of a - ib IS the acute angle MOP, taken negatively; for a rotation through this angle in the negative direction gives the same result as a rota- tion through the corresponding reflex angle in the positive direc- tion. 176. To find the sum of two complex numbers. Let a + ib and c + id be the given numbers. Draw the lines, OA, AB, BC, CD, representing the numbers, a, ib, c, id, in magnitude and direction, and join the ^ various :>oints as indicated in the figure. Then OE=a + c, ED = b + d, and OB = a + ib, BD = c + id Therefore OI) = OE + i.ED Geometrically Art 173 {a + c) + i{b-\-d). id IMAGINARY QUANTITIES. 153 Now, with the extended .neaning given to addition (Art. 173), OB + BD^OD. Therefore (. + ,7.) + (c + uT) = OB + BD =-{<^ + c) + i{h + d), 1. The result of combining the four numbers a ih c ,V/ ' independent of the order in which they are talen 1 t1 ' " bols obey the Commutative Law. ' '" *^*' '^"^■ 2. The numbers may be combined singly or in ^rroun. / .. obey the Distributive Law. ^ ^ ' •'•' ^^'^^ it Llf' r'""f '^ ^"btraction and the method of performing *um of the two numbers vould equal the suui or difference of :"r„:irte^7:t:r °^ - -" ^ -- *^« - - 154 HiailEH ALGEBRA. II But Thou IJ 178. To find thfi product of a cmnpUn numher, (/) htj a rml number; (;!') hi/ an imatjltuirt/ immlmr. 1. Lot it 1m> requiivd to multiply fk tlio cou»j)l draw PC parallel to JiA, meeting ^ OA i)rodueed in C; then from similar triangles, OAB, OCD, wo luivo OB'.OAxAB^OD'.OC'. CD. Kuc. YI. 4 OD^n.On, .'. 0C^7i.0A and CD^n.AJi. n{a + if,) -^ n . OJJ ^ OD = OC+CJ) Art. 172 = na + inh. 2. Let it 1)0 required to multiply the complex number a + ih by the imaginary number in. Turn OD through a right angle into position OE-, then OE represents i . n{a + ih) (Art. 1G2). Draw EF at right angles to CO produced; then the triangles ODC and EOF &ve geometric- ally o(iual, and CD = F and OC = FE. But considering direc- tion as well as length, OF^ - nh and FE-^ ina. Therefore t u {a + ih) -^i.OD-^OE = OF+FE Art. 172 == — nh + ina = ina — nh. Thus both those openitions obey tho Distributive Law. 179. The meaning attached to « + ih as a quantity, in connec- tion with the definition of Multiplication, determines the mean- ing of a + ih as a multiplier; for the quantity a + ih is formed by adiiing two lines, the first of which is draw.i in the direction l\ IMAGINARY QUANTITIES. 155 o tho or.g.Mal u.Mt and a times its length, nncl the second drawn at nghfc angles to ita extrenuty and b units in length. If then tins operation bo perfonned upon any lino (since any lino ^ay b^ considered a unit\ it is said to be nmltipHed by « + ih It w.ll bo observed that this operation turns tho lino multiplied 180. Tojind thei>roduct of two camphx numbers. Let it bo required to multiply a + ib by m + in. Draw the lino representing a + ii; then from the n.naning of multiplication by a complex number we have {m + in){a + .7,) = m{a + ih) + in{a + ib) Art. 179 = rna + imb + ina-nb Art. 178 = ma -nb + { (mb + n«), Art. 176 which proves tlio Distributive Law when both numbers are com- pex The student should draw the diagram corresponding to thi operation, when it will be found that an independent geo- metrical proof may easily be given. ^ ^ Similarly it may easily be shown that the Commutative Law IS applicable m this and the preceding cases of multiplication. 181 The modulus of the product of tu,o complex numbers u equal to the product of their moduli, and the aryun^nt of the j^o- d^txs equal to the Slim of thei^ arguments. From the product given in the previous Art. we have {ma - nhf + (rub + naf = ni\t^ - 2mnab + nW + m%^ - 2mnab + n'^b^ which proves the first part of the proposition. The second part s at once evident from the meaning assigned to multiplicatron hy a complex number (Art. 179). ^ ^^f^on 156 HIGHEH ALQEBIIA. 1 it ; 182. TIm modnluH of tJie quotient of two comph'.r, numbers is equal to the quotient of tlw.ir moduli, and the anjument of the quo- tient is equal to the difference of their anjmmnts. The truth of tliis proj)08iti«)ii follows from tho i)roco(15ng hy observing that tho product of tho divisor and <• X ami redum the remit U, tl«,f,ma I'+M then /• in .» tlie remit o/„d„tU,Uing a - ib. ' ^ For since P is real, it can involve only even powers of ib ■ and mce ,0 .s u„ag,nary, it ^„ i„voIve only «« powers of .7 The™ out J will remain unchanged. C«--If P=o and = 0, then ^ -(« + «) i, a «,ctor of the pvon function, and consequently . - (,;_ ,,) is also a ttor 187 It will now be instructive to briefly review the cour» of reasoning already given in connection with imaginary oultmet The meaning first aligned to the symbol •ri^niade" aTymW nght angle; then m connection with a numerical factor we made use as an operator. Having fixed its meaning, both as a "mW o operation and a symbol of quantity, we examined the Tuto of combining the quantities it represents with those repre entld by other symbols, and tr.aced the connection between i^T^ t.ons performed on the quantities themselves and the symS operations by which they might conveniently be represented J wT: T "' "'.'^ ""■^'' "' '^"■"S "" "' re- lished Id !*'• '""™*'°.'" <" ^''■"^"'^^y A'ge"- -- estab- iished, and smce imaginaries, both when taken alone and when combined with other quantities, have been shown to oW the fundamental laws of algebraic operations, the whole foZ one harmonious system and results obtained by the use of imagiiarie are quite as reliable as those obtained by any other nrSessIf mathematical investigation. ^ ^ ' "' 15a fimHER ALGfififtA. One point of interest still remains. The imaginary symbols, both of quantity and operation, are unintelligible except in con- nection with geometrical magnitudes. Suppose that in the solu- tion of a problem relating to other magnitudes the imaginary symbols are used, but that they do not appear in the result, is the result reliable ? To answer this we have only to observe that magnitudes of any kind may be represented by straight lines, and that by so doing the problem immediately becomes a geo- metrical one, and then all operations are intelligible. The result, when correctly interpreted, is therefore in such cases perfectly reliable. 188. We shall now investigate the properties of certain imag- inary quantities which are frequently employed in mathematical investigations. Suppose x^^J^ then aA^i or .r3-l=0, *hatis, (x-l)(x^ + x+l)=:0. Therefore, either ar-l=0 or x^ + x+l^O- 1± v~^ whence x=l or x = - Each of these values of x when cubed gives unity; therefore unity has three cube roots, namely, 1. l + V-S l-V~ 2 ' 2 "' two of which are imaginary expressions. Denote these by p and q; then, since p and q are the roots of the equation, x^ + x+l=0, their product is equal to unity. That is, pq^l, :. p^q=^p\ Similarly we may show that p == q\ IMAGFNAKY QUANTITIES. I59 189. The geometrical meaning of these results will l,e found interesting and instructive. Draw three lines, OA, OB, OC, each a unit in length, making angles of 120 degrees be- tween each pair. Join BC; then it may easily be shown that BC cuts AO produced at right angles, and that 0D=- and DB=^~ VH. We have, then. -o\ 2^ 2 -P* 2 2^' Now, since the absolute value of each of these expressions (the lengths of OB and OC) is unity, the absolute value of their product, or of any power of one of them, is unity; and since the sura of the angles A03 and (the reflex angle) AOC is 360 de- grees, we see that their product is represented by OA, that is, + 1. Again, y = 5r, because turning twice through an angle of l'>0 degrees gives an angle of 240 degrees; and q^=.p, because turn- ing twice through an angle cf 240 degrees gives a whole revolu- tion and 120 degrees besides. 190. Since each of the imaginary roots is the square of the others. It is usual to denote the cube roots of unity by 1, a, u? where Y -1; 3. Every nun.ber l-.as three cube roots. Let a denote the cube root of a nu.nber found in the ordinary way; then «o, and aj> are also cube roots. ^''^ (auty = aW = a^ It will bo observed that two of the cube roots are imaginary. 191. Wo shall now give a few examples: £x. i.— ]>ivide c + di by a + hi. c + di _{c + di){a~hi) a + hi. {a + fn){a-/n) = "^ + ^>'^+(f tc - hd) i u" + b^ _ ac + hd ac - hd , ~ + /y2 + a y- r= Therefore x 2 ' - 2 from which the required root is known. In this example observe : (1) In (3) the positive sign must be taken with the ra^lical because x and y being real, x"- + / is positive (2) The signs of x and y must be alike or different according as h IS positive or negative, since Ixy = i. -fi'a;. 5. — Find the square root of ± \^~7\, Assume Then therefore from which Therefore and ±^-l=^'-2/^±2.ryv/"rT, 3-2_y/2 = and 2ary=l, .r = y = ± a/2 ope^tC"* '''"■"'' ""^ *'"' ''"^™™ —ponding t» these r 162 HIGHER ALGEBRA. Ex. .^.—Resolve a-^ + y -^^^ ^.j^^g^ factors. We have a;^ + ^ = (.r + y)(.r2 -xy^ ,f). ^ow a> + o>=--l and 0,. 0)2=1, therefore ^ + '>f = {x + y){x + ^y){x + a,^). Similarly .r' - y^ ^ (^ _ ^^^^ _ ^^^^_ ^^^^ Art. 190 ^a,-. 5.— Resolve .r^ + v/^ + ^2 _ ^^ _ ^^ _ ^^ j^^^ factors. The expression may be written We have now to find two quantities whose sum is -{y + %) and whose product is f-yz + z\ Factoring this latter expression we get y + 0)^ and y + oPz; but the sum of these two expressions IS not - {y + z). If we multiply +he first factor by o> and the second by o.^, their product will be unchanged, and the factors become ••nj + ui'z and o>V + a>«, whose sum is -{y + z) as required. Therefore '^'' + y'' + z''-xy-yz-zx^{x + i^y + ioh){x + is,hj + t^), Ex. (5.— Factor a? + -t^-\.^- 3xyz. The expression may be written in either of the forms, a^ + M' + (<^'zy -3x.y + u>h and x + tohj + toz are factors; and since the expression is of but three dimensions there can be no other literal factor. The coefficient of ar' in the product of the three factors is the same as that of x' in the given expression; therefore '>^ + f + ^-^^yz = (x + y + z){x + ioy + u>-'z)(x + u,'y + u>z). The factors of the expression might evidently have been taken from those of the last example, and conversely. The methods of tniH example niigijt aiso have been used in Ex. 4. IMAmNARY QUANTITIES. jgg JiLltlu'^'^'T" "^ i-aginaries to the solution of geo- metrical problems does not fall within the scnnA nf +V, work Wfi mvo T.^ . P® **^ *"® present the square, on these three lines equals three times the suT of the squares on the sides of the triangle. i>i^rS^: perpendicular in Tin i i note the /.«,,/. of BC, CA, AB, J, L by . fl '^T' considering direction as well as length, ' ^' ' BA=.£D + J)A, CA=.CI) + I)A, sidf 7: t iTr ' ' ^^"^" ^? *'^ ^^"^^^ «^ *^« ^-^^^^ of the side, z.e It IS the square of the modulus when the side is ex pressed by a complex number; therefore = 2(a^-ax + x^ + y"). Again, FG=^FC+CG EL^EB + BL = -i.CB-i.CA =i.BC + i.BA ^ia-i{x~a + iy) ^ia + i{x + iy) = y + i{2a~x), =^-y + i{x + a), and Kir=EA,Aff^ -X- + ^.) -X- « + ..) = 2, - .'(2. - «). Then, as before, sum of squares on FG, EL, KH = {f + (2a - xf} + {f + (^ + «).j + 1 4^. ^ ^2^ _ which proves the proposition. NoTE.-In thoabovo draw fiChnriznnf^iw /.*!..„•-.. r. , tno usual directions for measnrpmnnt^ ^Zr"\V" "^'"' "^'^ ^pwarun; olioose between ,»«.„ a„d .»y«nZu8ho," °' """ '^'""^ ""ngulsh it 164 HmateR AL6EBRA. EXERCISE XVI. 1. Find tho values b' .•"+ .r'+ .,4+ ,.2+ j ^^j x'' + x'' + .t> + a; when X = 2i. 2. Simplify (2 - 3i){3 - 2i) + (4 - i Vsyi 3. Express as complex numbers, (2 - 3iy and (1 +t¥, 4. Simplify (2-^V5)3+(2+^,/5)«and{(2 + ^V5)2-(2 + ^V3>}(V'5+v/3). 5. Simplify ^. + 1:1?!' and -A-:l^l__ + Ji+ ^2 2-Sr 3->^ iV 3 - V2 V3~uVl' 6. Extract the square root of 5 - 1 2/, 1 - U V3 and ii VE -\. 7. Extract the cube root of - 3 v/ 3" -7; ^2 and- 10 + 9iV 3". 8. Find the values of «" + h^ + «'' _ 3ahc when ^ 9.^ Show by the use of imaginaries that {u^~3ahy +(30:^1 ^ly = {-(7^3+2v/5>'by 7-iv/5: 1 2. Simplify l±l^ + ^ + 3iVj_ _ 4(2-i vT) 2-iV3 2 + iV3 \-iV3 ' 13. Find the modulus of 3 + 4*, vi^~'n? + 2mni and ^". 1 - i* U. Express 69zI^;;Ti+( V 3 - 6 • S)i ■ 3 - ( V 3 liTS^^ '" *'■" '•'™'- « + •■'■■ 1 t IMAGINARY QUANTITIES. 155 15. Find the value of ^Yi^lU'^Y ^ 1 ^ , «^ + *^=0 and «.= 1. '^^"^ '^^ ■«^t>^T3-a-^.' ^^-n 16. Find the modulus of (?^^P + ^i) {Q + ii){l5 - Si)' 17. Find the p,-oduct of (a + ih){la + b){ia + ih), 18. Express (« + i6)(i + ,',)(, + i,^ ^, „ ^^^^^^^^ ^^^^^^^ 19. Show that What^relation exists between these quantities Before the, are 20 If ^ + ?i is a root of ax^+bx + c = Q, then ay + i;, + c = ««2 and 2«^ + . 0; a b and c denoting real quantitfes SW th'e necessity for the latter clause in this example. 21. Find the sum of 1 + 2/+ 3i= + . . . (^+ i)^,. ^j.,^ ^ j^ .j. even multiple of 2 ; (2) an odd multiple of 2. ^ ^M 1) an 22. Find the product of (a + 6 - ct)(h + c - ai)(c + a - bi). 23. Detect the fallacy in the following reasoning: (-1)^ = (-1)' = {(-1)^]^ = (+1)^ = 1, and Illustrate by reference to a geometrical diagram. 24. Find the modulus of 1 + i^ + i^,. + , . . ^^ -^^^ ^j^^^^ ^^ ^ 25. If .is an imaginary cube root of unity, then 1+. and 1+0) are the imaginary cube roots of - 1. 26 Show that (1 + co)^ and (2 + .)^ are cube roots of 1 and - 27 and hnd the other roots. ' 27. Find the values of (l+o>y + {l+u>y and (l-a> + o,»)(l+o>-o>^). 28. Simplify (u> + ^)(a>^-^•) and (1 + . - 0,=^)^ + ( 1 -a, + o>7 + (I _7. 166 HIGHER ALGEBRA. 29. Show that (1 - (0 + o>y = (1 + o) - w-)'" = ( _ 8)". OA T.1 1— o)l+a> a)+i ou. ii^xpress -— —- and with rational denominators. 31. Show that (1 - o> + o>2)(l - 0.2 + co4)(l - o,* + a>«) . . . . to 2.t factors = 2^ 32. Show that ^ , ^ and iu) are sixth roots of - 1. Find the other three, and illustrate geometrically. 33. Show that 2 V 2 ; -\^2"~T~) v/2 2 2 • ^ and give the geometrical meaning of these equalities, Show that the result will be unchanged by changing the con- necting signs of each of the factors, or by multiplying each of the second terms by o>; but if each be multiplied by t, the con- necting sign of the result will be changed. Give the geometrical meaning of each of these statements. 35. Simplify 1 — + + a + b + c a + huy+CM^ a + bur'+coi' 36. If x + ,j + z== - + 1 + 1 =0 and xy^= 1, show that .r, y, z are the cube roots of unity. 37. If ar + y + c= -+-^ + -==0, show that a;« + / + ;^ = and that A-" + f + z° = x'i/z{x^ + '/ + ;s«). 38. If x==a + h, ,j = auy+bw\ z = ao>''+bw, show that {1} x' + if + z' = Qab, (2) x'^ + ,f + ^= 3(„3 ^ j3->^ (3) x' + y* + ^*^i8,,^2^ (^^ x' + f + :^=l5ab(a'^+b% {5} xp==a'^ + P=-^,r + y)ri^ + ^)(;, + :r). IMAGINARY QUANTITIE 1G7 and that -Y^ + r» + ^2 _ j^^ _ ^ -^ _ ^^ then («^ + f,>/ + czf + (b, + ey + «.)« + (^^ + ,,^ ^ ,,^y ^ I ■ CHAPTER XII. QUADRATIC AND HIGHER EQUATIONS. ONE UNKNOWN QUANTITY. 193. The symbol denoting the unknown quantity in an equa- tion is frequently called a Variable. Symbols denoting other quantities are called Constants. It is often necessary to examine the result of assigning special values to one or more letters in an algebraical expression; the letters to which different ^^•llue3 are thus given are also called variables. 194. An Integral Equation is one in which the variable or unknown quantity does not appear in the denominator of a frac- tion, and is not affected by any root sign. It is in its simplest form when its terms are arranged in powers of the variable, and the coefficient of the highest power is unity and positive. 195. The Degree of an Equation is the number of dimen- sions in the highest power of tlie variable which occurs in the equation. The words, "linear," "quadratic," "cubic," "biquad- ratic," are used to denote equations of the first, second, third and fourth degrees respectively. These terms are especially applied to integral equations. 196. From an equation which is irrational, or fractional, or both, an integral equation can be derived, the roots of which are usually assumed to be the roots of the original equation. Upon QUADRATIC AND HIOHER EQUATIONS. 169 trial however, it is frequently found that one or more roots of one equa ion will not apparently satisfy the others. This point deserves the most careful consideration. ^ 197. The following method of rationalizing surd equations is mstructive Arrange all the terms on one side and denote them one term '''^'''"^^ quantities, if any, being collected into (1) Let there be two terms. Then the equation is a + b = 0. Therefore (a - b)(a + b) = 0, which will be rational, since each term is a square. (2) Let there be three terms. Then the equation is a + b + c = 0. Therefore ia + b-c){a-b + c)(a-b-c)(a + b + c) = 0, or a* + 6* + c* - 2a^^ - W w VV* 0) (2) Then Factoring, Therefore that is, Let If we let QUADRATIC AND HIGHER EQUATIONS. y' + 2y=80, or y' + 2y 80 q. (y+l0)(y-8) = 0. y= -lOo-c 2-= - 10 or 8. 2* = 8 = 23. 2'= - 10, the solution cannot be obtained. Ex. 5._Solve (a: + «)(^ + 2a)(ar + 3a)(a: + 4a) = c*. Multiply together first and fourth fn.f third factors, since the sum of a and 4a T.^^ '"'"' ^"^' of 2a and 3a. ^" '' *^^ «^™« «« the sum Then (^+5aa,- + 4a')(^2+5aa. + 6a==) = o«. Let Then or Completing square. Extracting root, or »r;rr-r:xr.i---;;-...., or a;' + 5aar + 4a2= -a^ii/^Tj:^^ ^' + 5aar= -5a2iV'^*T^. This equation, althoimh n„w,u.-— ^ «. solution. " ~° """^""'' "°^^'^ "« difficulty in its p 172 HIGHER ALGEBRA. 'ff Ex. ^.— Solve ^2 + a.-2 + a; + a.-i = 4. Add 2 to each sidf of the equation. {x + x-^ + 3)(a; + a:-i - 2) = 0. ar + a;-^-2 = 0. Solving in turn a: + a;-* + 3 = and a; + a;"^ - 2 = we find Then or or Therefore or (1) (2) x = x= 1, 1. / y Ex. J.— Solve a^+2x^- S.r' - 3^;^ + 2^ + 1 = 0. Arrange as follows : {a?+\) + 2x{x'+\)-Zx\x+\) = 0. (1) It is evident that .r + 1 is a factor of each quantity in brackets, .'. a: = - 1 is one solution. Dividing (1) by {x+V) we obtain {x^-a? + x'-x+l) + 1x{x'-x+\)-2>x'' = 0, or a;* + a;^ - 4.t' + a; + 1 = 0, or (a;*+l) + a'(a;'+l) = 4a;'. Adding 2x'^ to each side of the equation, (a;* + 2a;2 + 1) + x{x'' + 1) = Qx\ or {^x-'^\y + x{x''+l) + j = ^. Extracting square root, , _ a; 6x ^+^-^2=±T- TI or or 0) (2) QUADRATIC AND HIGHER EQUATIONS. I73 We have now two quadratics to be solved, viz.: X a:'+l + J= _ 5x and (1) its, 2 ^2' ^ neither of which presents any difficulty. -sa.ew.entheter--frX^^^ Reciprocal equations may also be defined as those which are not altered by changing aj into - . Every reciprocal equ^^^ Joda degree will be divisible by IS -lorl .^"? "^" '' ''™">' ^^^^^^-^^ - *^e ^-^ tern' if. 1 \ . "^ ^' ^"^ ^^'^^y reciprocal equation of even degree with 2nTZ '■ "'^ '^ '^'"^^ ^^ ^^ - ^ ^ -^ *'- reduced eq a degree, and with its last term + 1 (Cdenso's Algebra). one of thVr^r ^"'Tf ^ "'^ '^ ^^'"^^^ '^ ^ ^-^-^-» -d L n Ex 4 " ""'' '^^"^ *^ ^ '^^--^raHc, and then s Ived J'ar. ^._Solve-?-t£--a Add 1 to each side. Then (l + xy+l + x* or or 2(i+x+xy (1+^)* ~ 2"' 174 HiQHER ALGEBRA. Extracting square root, l +x + x^ ^ Vj(l+a) = ± or or 1 + x + x^ _ V2(l+ a) X or 1 + 2.r + a:^ 1 X 1-m' This now can be solved as a quadratic. I Ex. 7.— Solve v'il+xf- 7(T3^2= vT^l^, 2 2 1 that is, (1 + ar)"* - ( 1 - a-)"» = (1 - x^. Dividing by (1-4 (l±fy'_i.(i±fy'/ 2 /l+a-\"* Assume Then or Solving for y we get that is, 1 V5 l+.ry 1±V5" /l4-.ry» • 1+a? /l±vT\ ^(liVT)™ ~ 2'" or diffttt! """"'"'°" ""' O-n.i.ator, .„a dividing b, their a? = (l±y 5 )« - 2"* \2x~3) +i2^3J ^Isii^FZ^)- Now, ... (?^^)^/?iz3^^ 4J2.+3 2.-31 V2x-3y +U. + 3; -i3t2J=3 + 2^3}- 2x + 3\i 2-^+3 2x-3_2(4ar'»+9) 2:^-3 2.r+3 ~J^^:rg~* Assume Then /2x + 3\* It is evident that y+l ig a f«r>f«^ «# i. i.u • , 1 ^ 2^ + y »« a factor of both sides, ar^d therefore y + - = will give a partial solution. If or 1 y+-=0, then y2=-l, Cubing /2^+3\S U^-3/ = ~ /2^+3\2 W-s) "^ ~ 1. or To obtain the remaining solutbr.s ^e have or 3^- 4 176 Solving, that is, or HIGHER ALGEBRA. y = 2, ^. -2. -|, r2x+3\i /2x+3\4 1 l2^::ij =2, or-, or ~ 1, or ~ - 2ar + 3. 1 2^33 = 8. or-, or -8, or 27 7 1 2' 1 8* ^a;. P.— Solve Vx^ + ax -\+ Vx^ + bx -\=. V a + V b. (1) Now, (a:2 + aar- 1) - {:i^+bx - l) = ^a - 6), that is, ( Vx'-^ax~\y - ( V'ar2 + />.r - 1)2 = .r(a - J), (2) Dividing (2) by (1), Va^ + ax-\- Var-^bx-\=x{ /a - -/Z) (3) Adding (1) and (3) we obtain Let 2Vx^ + ax-\= \/a + Vb +x{Va- V b). '^a+ V b =m and V^ - \^'b=n. li ,*. 2 A/a;* + aar - 1 = m H- wa-. Squaring, ix^ + 4,ax-4: = m^ + 2mwa: + n^o-'. Transposing, and arranging according to powers of ar, ar2(4 - a;2) + ar(4a - %nn) - (4 + w;2) ^ q. But >n^^ = (\/a-^. 'v/i'X'/a- 'v//r) = a--J, .-. ir2(4-w2) + ar(4a-2a4-2^>)-(m2 + 4) = 0, 0"^ ^'(4-n2) + .T(2a + 2i)-(m2 + 4) = 0. Factoring, {ar(4 - w^) + (m^ + 4) } (a: - 1 ) = {since m^ + 4 + w" - 4 = m" + n" = (^« + ^4)' + (Va - V'l)2 = 2a + 2A|. (4) QtTADRATiC AND HlOHEB ISQUATIONS. From (4 ) we obtain ar = 1 or -!±! that i IS. x^i or i:^C^±:^)!±i (Va- V bf~i Assume « ^ i . a-x = m and ar-6 = n. .*. m + n = a~b, • ^!!±!L*_4i/ 41 •-< 20(^* + n^) = 41 (m^ + ^^)(^,2 + ^^ ^ 2mn), 21(m* + n*) + 82m'n' + 82mn(m' + n^) = o. Arranging according to powers of ^., 21m* + 82w«n + 82mV + 82mw« + 21 w* = 0. Dividing by mV, 21^'+ 82^* + 82 + 82^ + 21^^! = 0. or or Let Then n 2V + 822/ + 82 + ?? + 2i = y 2/' 177 (1) throwing (Sthe L'' ^ ^^™^'^ ^°^"^^^^ ^^ ^^^^^^ ^^ 21K + n'^)3 + 82mn(w« + ,,2)^40^,^,^Q^ * ''^' ^"^^ "t^ reaauy factored. 178 HIGHER ALGEBRA. Ex. ii.— Sol ve 2a^ -a?~ 2.r + 1 = 0. Add ar* to both sides. Then i>^+2o^-x^ -2x+\=x\ Extracting square root, x^ + x-\=z±x^ :. ar-l=0, or 2x'^ + x-\=0, :. x^\, or (2ar-l)(a:+l) = 0. Therefore roots are 1, -. - 1 ' 2 Ex. 7^.— Solve 2x* - 4ar + 1 = 0. Multiplying by 2, 4:r* - 8ar + 2 = 0, ®^' 4aj« + 8ar2 + 4-(8x2 + 8a- + 2) = 0, ^^ (2ar2 + 2)2-(2i/2.^+ V 2)2 = 0. Factoring, (2a^ + 2 - 2 A/ 2 . :r- V 2 )(2r' + 2 + 2 V' 2". :r + 1/ 2 ) = 0. .-. 2r'+2-2v/'2.ar- i/'2=0, 2a;2 + 2 + 2v/2".a?+ V'2" = 0. V 2"± ^2 v^Ti or From (1) we get From (2) we get x> x = 'V/2±v'_2v'2-2 r2> Ex. 13. — Solve x+ Vx+\ = 5. Rearranging, V'^+l = 5-3-. Squaring, ar+ 1 = 25 - lOar + a^. From which a- = 3 or 8. Upon trial we find that 3 satisfies the given equation, but that o belongs to the equation, X - VxTl = 5. QUADRATIC AND HIGHER EQUATIONS. Ex. i^.— Solve x+ '^t7Z\^=,\ 179 ^a:. 75.-SoIve ^2FT1- '/57Tl-7 = o. Using the result in (2) Art IQ? , u when a, ^ and c are the tts o tv ^^ '""^ ''" "'^"' result in the following fori ' '^"''^''^- ^^^^"^^ *his a V - 2fi2 - 2c2) + (42 _ c2)2 == 0. Here taking ^s^^g A2-9^.o , ^ the expression reduces to 9a:*- 698a: +2013 = 0, (^-3)(9a:-671) = 0. :. x = 3 or 74^. l^pon trial it is found that apparently th. « • • , not satisfied hy either root and tlf.K ^^'"^^ equation is equation, ''*' ^'^^ *^^* *^« root 3 belongs to the (2) or (1) (3) 1/ 2a: + 3 + ^ 5^~jri; _ 7 ^ Q^ and the root 74f to the equation, The difficulty may be explained in two ways: (a) It has already been explained in Part T fhn, .u • he square root of a quantity may be either J'-! "'^ "^ If we take positive signs of the roots of the r^'*"' T '''^''''''' oigii Or tne roots in V fla* j. i o„j xl . . a 1X1 r oa: + 1, and the positive fT^ff/jg^i 180 HIGHER ALGEBRA. sign in •/ 2^; + 3, the equation will be satisfied by a; = 3. A similar explanation applies to the root 74S. (6) The difficulty may also be explained by pointing out that equation (1) is the product of four factors, of which (2) and (3) are two, and consequently any value which satisfies either (2) or (3) must satisfy (1). A similar explanation applies to Exs. 13 and 14. If we are restricted to the positive root in each case, the given I equation has no solution. Ux. 16. — An integral equation of the third degree can always Let '1 be expressed in the form a Substituting, and arranging in powers of y, we get 2(? ah .^i! Now, if this equation can be solved we shall have the roots of a the original equation ; for x = y—-, and therefore when x is o known y is known. If, therefore, we can solve a cubic equation in which the term containing x"^ is wanting, we can solve any cubic equation. > , / / » ^^y^-^Ex. 17. — Solve the equation, a^ — qx—r — 0. Let Then ^ = 2/^ + H^-^ 3y t''+ ,3 (-1;) '21f It ^.'-f or 3^ —qx — '^f ■\- ^AAM^ 272/= = r. I or ' ! ? Then QUADRATIC AND HIGHEK EQUATIONS. 181 from which and f-- x = y + 3y -{l-^l-^P{l-4U}' after rationalizing the denominator and simplifying the second term. It should be observed that the same result is obtained by taking either the upper or the lower signs with the radical. But every quantity has three cube roots, and we must determine which are admissible. For brevity, denote the first term by p and the second by q, and the cube root of unity by ■\"/-,ora,\^/,^.+a,, .>2 3 •^ -^< /'■\~w 10 B (X Air/ Lv«l*yt'" f(jA/ ^/ ^^«*^' tiny U /-^ ")( 182 ' A^ :vf' AAj Vi%/iti(^ HIGHER ALGEBRA* Ui t^*"*'^ ^A/v^ Jtf^ EXERCISE XVII. Solve the following : - a {x - h)(x - c) Hx-c)i^_-^ _ ^ \a-b){a-c) "^ (b-c){b-.a) '"'' .-.. '' a- {a-xf + {x-hy 5 (a - x){x - b) X r 7.5 / 1 6. C8. x'-^x-lb x''+2x-Zb x'+lQx + ^V 1 1 lx'-Ux+2 i^x'-lbx^ 2 8 8 a^ = 12a:2_7a;+l. + 6a; + 5 x^-Ux + ^b x^-lQx+^' 6 5 + x^-1x-\-\0 x'-l'dx + iO' {a - xf + (g - x){x - &) + (a? --&)'^ =^9/ 11 1 I ^9. (.r-7)(a;-3)(a: + 5)(ar+l) = 168#. ^ ^10. 16ar(a; + l)(a;+2)(a; + 3) = 9. ,^11. Va^-a''-b''+ Va^-b'^-c'- \/ x" - n, ^^y c'lJ^-^--. IMAGE EVALUATION TEST TARGET (MT-S) 1.0 I.I U& 12.2 2.0 1.8 L25 i 1.4 liiii 1.6 /2 >> Photographic Sciences Corporation 23 WEST MAIN STREET W2BSTER,N.Y. 14580 (716) 872-4503 l\ iV :\ \ ^\^ ri^ "«t:^ 5q^ 1 +4a;- 8a;2+2ar* = 0. 51. (5:r2+^+io)=+(a;2+7a:+l)2 = (3a:2-a;+5)H(4a:2 + 5^ + 8)l / 52. (12:r-l)(6ar-l)(4ar-l)(3ar-l) = l. • 53. 8^+ 81 = 18^:^460?*. 55. (l+a:8) + (l+a;)«=»2(l+;ir + .r2)*. 56. a?*-2a:"-3ar»-12a:H-36 = 0. 57. x*-8ar^+10x^ + 2ix + 5 = 0. 64. a;' + 3ar = a'- »■ ,3* '^ ' /68. {x + b + c)(x + c + a)(x + a + b)^{x + a)(x + 2b){x + 3c). 4 69 ^0 20 8__ 12 _ . / * a:' + 2a;-48~ar2+9ar + 8 aT2+10ar'''^T5^^^'*"^"^' ^ 60 (^ - ^)(^ - c)«'' {x - c){x - a)b^ (x - a)(x - by , / * {a-b){a-c) (b-c){b-a) "^ {c-a){c-b) ~ ' ,61. (a!3-2a^-2ar + 3)(a;3-4ar» + 4ar-3) = (ar» + 2a:'»-2.'K-3)(ar3 + 4a;2+4a;+3). ^ /TO 2\3 62. «* + 2a3? + ^-^— = fby putting it in the form \or-^ax 63. a;*-8a:3_208 = 0. 4 65. ar»-18ar-35 = 0. / 67. ar»-15a:2_33^^g^7^Q 69. 8a:3_3g^^27 = 0.^ 64. a:* --10ar»- 3456 = 0. 66. «'+ 72a;- 1720 = 0. >^ 68. 2ar' + 25a:' + 56a; -147=0. CHAPTER XIII. SIMULTANEOUS EQUATIONS OF THE SECOND AND HIGHER DEGREES. TWO OR MORE UNKNOWNS. given for the solutinn „f , °''""^™ *"»* »<> general rule can be s ."' '™ '"'""on of auch equations. A carofnl •*,.j . xi folWmg examples will enable him to solvt I^^l" ^°' ""^ tent and useful problems that ma/a^ " "■" """"' "^^' method, not introduced "plrt 7 of I^ -me explanation of a ^ of the >. de«r.. ^ mircKh^TrinT terminate Multiplier., and is best explained i a' ilU^ut Bx — Solve 0) (2) (3) ing^he ^fulT T" ''' '' ' ^^^ -' -^ -^^^^^ <^)' -^ -ng- 18a HIQHER ALGEBRA. Now let / and m have such values that the coefficients of y and z may both be zero. Then ^_ld+mdi + d. where and Z=i — ___ la +mai + a^* Ic + mCi + c^axO. m 1 ftjC, - b^i 6jc - Acj bci - b^c ' Substituting these values of / and m in Id + mdi + dj la + moi + ttj we obtain x = «(*iC2 - Vi) + ai(V - bcj) + ^{bci - 6jc)' Haying thus found the value of x, the values of y and « can be written down by symmetry. It is evident that this method may be employed when more than three unknown quantities are given, all that is necessary being a corresponding increase in the number of indeterminate multipUers. The student may, as an exercise on this method, take any ordinary selection of problems in simultaneous equa- tions, and find the solutions required. 201. We proceed now with the consideration of the subject, matter proper of this chapter. 202. When both equations are homogeneous and of the same number of dimensions, the method of elimination may be em- ployed. -£'«.— Sol ve j/'-3xf + bx'^y = 1 5, xy^-ix'y-^^d^^b, Multiplying (2) by 3 and subtracting from (1) we get (1) (2) can EQUATIONS OF THE SECOND AND HIGHER DEGREES. 187 Factoring, (y - x){y - 2x){y - 3a-) = 0. ^^"^^^'^ y==x,2x or 3;r. Substituting these values successively in (1) we get Then and «'=5, 2 ^^ ^• v^5 y= v'S; ^20 or 3. Each pair of roots may be multiplied by an imaginary oube root of unity, giving six other solutions. 203. It is sometimes convenient to find the values of ar + « and xy btfore finding the value of each letter separately. -£a;.--Solve x^ + f + x + y=\Q^ &{x + y) = bxy. Equation (1) may be written (^ + y)' + (a; + y)-2a:y=18. Substituting for xy from (2), (1) (2) {x + yy--{x + y)-\S=^0, from which a; + y = 5 or -— ■ 5 then from (2), Squaring (4) and subtracting four times (5), xy = ^ or -— . Combining (4) and (6), «-y=±l or ±- 1/2I. 3 a: = 3, 2 or -(-3±i/21), (3) (4) (5) (6) 3 y=2, 3 or =(-3qFV'21). 188 HIOHER alqebha. When the values otx + y and xy have been obtained, the values of the separate let^^rs may be neatly written from the quadratic whose second term is th:. sum of x and y with tlm sir/n cJianged, and whose product is the last term, a different letter being used as variable. Thus from the preceding example we have r»-5r + 6 = and r^ + i? r- — = 0. 6 25 ' from which r = 2 or 3, or '-5(-3±V2?.). The two values of r derived from either equation will give two solutions, one value being given to x, and the other to y. 204. When one solution of a pair of simultaneous equations has been found, other solutions may frequently be written at once from the following considerations : 1. If the variables are symmetrically involved, their values may be interchanged. (See Art. 203.) 2. If each term is of an even number of dimensions, the signs of both values may be changed. 3. If each exponent is even, the sign of each value may be separately cht nged. 4. If the literal part of each equation changes signs when the variables are ' iterchanged, and if each term is of an odd number of dimensions, the values may be interchanged, providing both the signs are also changed. 205.— Another artifice sometimes used is the finding of the values oix + yB,nAx-y before finding the values of x and y. Ex, — Solve Assume (a; + y)(a:3 + 2r')==76, {x + yf=U{x-y). x + y = 7n and x — y = n, m+n m~n .. x = — — - and y = — -— . (1) (2) EQUATIONS OF THE SECOND AND BiGHfiR DJlOREES. 189 Then o^+f = {x + y){x^-xy + f) = ^(^' + -4—) 4 • Equations (1) and (2) now become »n»(3n2 + m2) = 304, and Dividing (3) by (4), w'=64n. 3w» + m2 19 or But from (4), or or Let wi 4w* 4w<3w2 + m2) = 197». 64' . mV3m« \ •• 16164^ + ^7 = ^9^ 3m* m* 16 X 642"^ 16 3w8 m le'^x iP^le + 1-T=19. 44 '**• Then (6) becomes or 3«2+i6«=i9, 3«2 + i62_l9:=0 = (3«+19)(«-l) = o. .*. a=l or -~. Solutions can now be readily obtained. (3) (4) (5) (6) 1^ BmHER ALGEBRA. 206. The following are further illustrations of methods: Ex. 1.— Solve a^ = 31ar»-4y», (\\ Dividing (1) by (2), Let y» = 31y'-4ar». ocr'_ 3lx-- 4f 3/»~31y2-4^' X - =711. y (2) (3) Then m ,8. 31w2 - 4 or or or 31-4m«' 31m'-4w« = 31«i2-4, 31(m»-m«) = 4(m«-l), 31w2(^-l) = 4(m-l)(m* + m3 + m2 + m + l). (7) (4) (6) (6) It is evident the equation (7) is satisfied by m= 1, or - = 1, orar = y. If x = y, then ar = 3/ = 27. ^ The equation left after w - 1 is struck out is 31 w' = 4(m* + wN- w* + m + 1), 1 1 or or or Let Then 31=4(m2 + m+l+- + — ) 31=4(. + J,)^4(.-.1)^4, 35 = 4(m + iy+4(m + iV m + — = «. 35-4«2h.4«, from which we obtain 2«+l=±6, or 2« = 5 or -7. 0) (2) (3) (4) (5) (6) (7) = 1. EQUATIONS OP THE SECOND AND BIOHER DEGREES. 191 1 . If m + - is now substituted for «, and than - for m, we obtain X in terms of y, and find the values of a: and y to be Other values are. 2(3±v/33) and | (3^^/33). ar=15, 30,1 y=30, 15J Ex. ^.— Solve (ar + y)* + (a- - y)* = a*, (x» + y2)*+(x»-y«)*=a*. Cubing (1), a; + y + ar-y + 3(a^-y»)*a*=a, O' 2ar + 3(a;«-y*)*a*n=.a, ®' 3(a:2-y'»)M=a-2af, (1) (2) or /«a «\J a - 2a? (a!«-y2)»= 3a* (3) Cubing (2) and treating the equation in a similar fashion to (1), -..4a_ «'-2^ (^-/) 3a i Dividing (4) by (3), ^ ^' a-2x J* Substituting these values in (2) we get a" a-2x a^~2x' ~ + 3a* J (a - 2a?) -^-J. (4) (5) (6) (6) can now be readily solved as an ordinary quadratic. The values of a; and y are: lofi filOfiDR ALClEBftA. Mx, S. —Solve a^^az + by, Subtracting (2) from (1), ^-y'-(«-A)(^-y). <1) (3) The equation is satisfied by putting x-y = or x=y. If a^=ax + bx, and x = or a + b; .'. y = or a + b. Dividing (3) by (x - y) we get x + y-^a-b or y = a-b-x. Substituting this value of y in (1) or (2) we find ^ = 2 {(« - *)± V^(«-A)(a + 36)} and therefore y = ^ {(a - i):|: V" (a - b){a + Sb)} . Ex, 4.— Solve «♦ + s^ = 706, a? + y=8. ^'•om(2), (a: + y)* = 4096, . or «* + 4ar«y + 6ar2y2 + 4ary3 + 2,« = 4096. a?* + y* = 706, /. 4r> + Q3»f + 4a-y3 = 3390, 23:^2/ + 3;ry +2^=1695, a'y(2jr2 + 3;ry + 2y) = 1695. 2ar2 + 3xy + V = % + 2/)''-a»y «=128-ary. But or OP Now, 0) (2) 0) (3) from (2) \P^ (1) <2) (3) fcOUATlONS OP THK SECOND AND HiOfiJSft DEORUES. 163 Substituting in (3), a^(l 28 - xy) = 1 695, 0' ar«y»-128xy+ 1695 = 0, ' (4) or (ar2/-15)(:ry^ll3) = 0; /. ary=15 or 113. Combining these results with x + ij=8 we readily find the values of x and y. One set of values is ar = 3, 5,1 y = 5, 3./ (1) (2) <1) (3) (2) EXERCISE XVIII. Solve the following equations : 1. r' + y'=65, ar + y = 5. 3. a:5-/= 16564, x-y = i. 5. a!3 + 3/' = 35, ^- a:' + y'=13. 7. a;* + ar2y2 + y«=133, ar' + ary + 2/^= 19. 9. a:» + ary + y = 37, i/^ + xy + x=l9. f4K 11- (^ + 3/)(^ + 2/') = 1216, '^ '^"^^^ ar2 + ary + y2 = 49. 13. C^+y)' , (^-y)' _ a" 62 wi. 1 «^+y''=2'*- 2. a!^ + y* = 2417, a: + y = 9. 4. a^ + y* = 641, 6. a^-y* = 6&, '^ ^iH'*^ x-y=\. 8 ar' + ary + ya^gi, ar+ Vxy + y =13. 10. a^-i :y=a«, a^ + 2xy + 2f=h\ 12. ar« + y' = a', x'y + xy^ = h% 14. a' i2 ma^I? {x+yy {x-yf {x'-yY 104 BIQHER ALGEBRA. 46. :i' = 'i/', ^ 17. x»-y, / 19. ar(x« + y2) = 6y, ^16. x»^ = a», / -i -i 5 20. 5_y=,.i+y. y X (B»fy«' ^21. ar + ary»=18, ^ -■ I t 22. ar» + ar\^ary»=208, > y' + y^^= 1053. a:y+'ary»= 12. ^^* Vy"^0(^^l)^^^' C24. :r + y = ;ry = ar»-y». ^ S-')(^*^)-"»- 25. (ar» + y«)(r» + 2r') = 455^ 26. x* + i/^ = 6U, y « + y = 5. ar»y + ;rys=290. 27. ar-y = a, 29. a?» + y = 7, / y« + ar=ll. r ^28. (Find 16 solutions.) 1-xy «— = 2, / = 2. 30. ^ + y ^ w»+w t» * a; + y x-y n*-m^' _g_ _y n? + 2ran-m^ x-y x + y~ n^-m^ 31. aJ' + y6=178\/3. / ^^32. af'+«'» = y*, y x' + y'=lOxy. y"+'" = ar*. \^ < I BQtTATlONS OF THE 8EC0ND AND moBlB DEGREES. 195 rt + ar 6+y a + A + o* aJ+y-ft 36. -^^^' + iv:j:^/^„^j^ 3g ^y,4^4^J^_ 1^ 37. ^ + y> + :^(^+J,)>I3, ^ 38. ^S;+^^ = , + j 19. V^ar + y+^ ,_y^4 a?»-y»=,9. ^M^^y')! = ^. (^-A^T- a; 3' / X 15 y 40. 2;* « mar 4- ny, y* = my + n.^ 42. (ar»+y')(ar + y) = l5;py, ^ (^+y*)(ar» + y») = 85xV='. ^^4. y"'+llx«y = 480, ^ 9^2 ar' + ay = 30. Vy Var 2' 45. ^(^ + aW + A^) + ^(^M:^^^^^??^ = (a + J). ■£"«. 1. — Solve a: + y + «=4, ar« + y2 + s2=14, ^y+yz-zx=6. 0/ (2) (3) (4J (5) '^^ HIGHER ALGfiSRA. Squar J (1) and subtract (2) from it, and divide by 2, Combining (3) and (4), 2/(^ + 2) = 3, zx= -2. Substituting in (5), 2/(4 - y) = 3, ^^ 2r'-4y + 3 = 0, from which y = 3orl. M 2/ = 3, ar + «=l, and from (5), zx=^ -2. Therefore ar = 2 or - 1 and ;== - 1 or 2. Similarly, when 2/= 1, ar= 1(3± v/l7), z^^hs^vVf). From the above we see that y has two values, whilst x and ;. have each four values, and it is necessary that these values should be correctly grouped to give a true solution. The correct arrange- ment is : ° x-^ 2, -1, l(3±Vi7); y= 3, 3, 1; 2=-l, 2, 1(3 T ^17). ^x. ^.— Solve X1/ + 1JZ + zx = a' - ar' = i2 _ 2/2 = c' - «2. Rearranging and factoring we get (x + y)(x + z) = a^, (y + «)(2/ + -^) = i', {!S + x)(z + y) = c\ 7 EQUATIONS OF THE SECOND AND HIGHER DEGREES. 197 Multiplying the three equations together and taking the square root we get , {^ + l/){l/ + «)(z + x) = ±abc. Dividing this result by each of the original equations in succes- sion we get - + » = ±^, 2,H-.-±l", . + .>±!^. « c Adding these equations and dividing by 2 we get x + i/ + z = ± Thence by subtraction, x = ± 2abc 2abc ' from which the values of y and z may be written from symmetry. I!x. 5.— Solve x + ij + z=10, xy+yz + zx = z\, xyz=:ZO. Equation (2) may be written ^i/ + z{x + i/) = Sl. From (1) and (3), ^,+^=10-;., xy = ^, z Substituting these values in (4), 30 — + 2;(10-;s) = 31. Rearranging, x^ -10z'^ + 31z- 30 = 0. Factoring, (z - 2)(z - S)(z - 5) = 0, from which , _ o q The values of x and y are then easily found. (1) (2) (3) (4) (5) (6) \ji 5. 198 HIGHER ALGEBRA. The following solution is worthy of attention : Consider the following equations in r: (r-z){r-y){r-z) = 0, or ^-{^ + 2/+zy + {zi/ + i/z + zx)r-xyz=0, Its three roots are evidently x, y and z. If, then, for x + y+z, xy+yz+zx and xyz we substitute their values from the original equations, we shall obtain a cubic equation in r whose roots are the values of ar, y and z. The equation will be ?^-10r2 + 31r- 30 = 0, which is identical with (6) previously obtained. The three values of r, viz., 2, 3 and 5, may be assigned U)x,y and z in six different ways as follows : a:=2, 2, 3, 3, 5, 5; y=3, 6, 2, 6, 2, 3; z = ^, 3, 5, 2, 3, 2. The methods of this solution are of great importance in the application of Algebra to Geometry. Ex. ^— Solve {x - y)(y -z) = a\ {y-zXz + x) = b\ (z + x)(x + y) = c\ Divide (1) by (2) and (2) by (3), and simplify. (b^-a^-b^y-ah^O, Px + (b^~c')y + c'z==0. From (4) ind (5), •^ _ y X _ x-y y-z 2a\b''-c^) -26*' (1) (2) (3) (*) (6) T D or El Eli («) I EQUATIONS OP THE SFonwr* *i.t^ „ iilB SECOND AND HIGHER DEQREE& 199 Substituting from (6) in (1), from which then from (6), and £x. 6. — Solve x=. a*A'-dV-cV 2 = 26V^Tr32 Adding the equations and factoring, Multiplying the three equations and dividing by (4), Then "'"'"■'^<'>'(^)'""»(3)''y..ya„d.,a„d^ding, «' A3 c3 r+-+-=o, Eliminating x from (5) and (6), Eliminating a? from (1) and (5), J (1) (2) (3) (4) («) (6) (7) (8) 200 HIGHER ALGEBRA. From (7) and (8), from which y = \R+^) ** Substituting from (D) in (8), a\E-by ' b^B-c'y ' Ji^(a^ + b%R + b^) from which Then r»= - ar3= - and ,f^- 3\2 \E-a^) (9) The values of a^ and 2/* are written from that of z' by symmetry. 208. In the application of Algebra to Geometry symmetrical expressions with three letters frequently occur. They usually arise from the sides or the angles of a triangle, or from the three dimensions of space. A knowledge of the more usual forms and facility in making transformations is desirable. The solution of the following equations will furnish exercise in such work. The following identities will sometimes be found useful: ^•^+y^ + z^--={x + it + zf-2{x9/ + yz + zx). x^ + if + z^ = {x + y + zf-3{x + y + z){xij+yz + zx) + Sxyz. ^2(y + z) + yHz + a;) + z\x + y)^{x + y + z){xy + yz+ zx) - Zxyz EQUATIONS OF THE SECOND AND HIGHER DEGREES. 201 EXERCISE XIX. Solve the equations: q1. a; + y + « = 6, Cl2. ^3. a; + i/-z=l, ^4. ary=15. ^ 6. x + 9j + z = 0, c^6. xy + yz + zx = - 7, ^7. a; + yf2; = 5, 8. xyz = ^(i. ^^ ^ 0. x\y + z) + i/'{z + x) + z\x + y) = 22 x + y + z^i, « + y + « = 8, x^ + y^ + z^=90, yz + zz— xy=i3; 2x + 3y-z=6, xz= - 2. a; + y + «=2, ^y + y^ + 2!^ = — 23, a?y« = - 60. x + y + z= - 1, ;r3 + 2/3+.3_ _97^ i^lO. t 12. <14. 16. x + y + z = 9, xy + yz + zx='26, {y + z)(z + x){x + y)==2l0. xy + x + y = 7, yz + y + z= -9, zx + z + x= - 17. {x-y){y + z) = a\ / (y + ;s)(2-a;) = i^ ^3. ^Pa^vXi-L. \'<^J x^ + xy + xz^ 18. y^ + yz + yx= -30, »* + «a; + «2/ = 48. y — y» + 2;=l— a, z — zx + x=\ — b, x — xy-\-y=\ — c. (y + «^ a;)(y -"« + «) = 6^ ^ {z-\-X;^y){z-x + y)==c\ x^-yz = a^jjf \ ^7. y ~aX = o, ■p,\----.u i • - / s? -xy = c\ / x'^ + 2yz = aPj ,.2 , 0~^ ,2 ,';2 + 2ar2^ = 6^. 14 962 HIQHEB ALGEBRA. If { 18. z^ + xy + i/ = 37, 19. y' + ya + «'=19, i? + zx + x' = 2S, >20. -^ «y(af-2/) = 12, c 21. -^ y«(y-2)=-30, ^. *ar(« - a?) := 48. / %!-«")= -30, y(«'-^)=-i2, , 23. Y^^vi 2/«(.-^)=-16, / y\x->ry-\-z)-\:^-\-x^-\-zx) = h\ W z(x + y + z)-{x^ + y^ + xy)=<^. ^ yz{y-z) = h\ . Zx(z -'X) = (?. ' f-zx^hy, s?-'Xy = cz, i ^^J>ia<3 25. c ^7. {x-\-y){x-{-z) = ax^ {i/ + z)(y + x) = by, (z + x)(z + y) = cz. !>^ + y^ + s^== 3xyz, x — a = y — h = z — c. 26. 28. 29. x^{y + z)^a\ y\z + x) = h\ xyz=^ (« + ar)(ar -y)=:ax. iQ/ . 481 13(a; + y + «) = —, x+y + z = 10, yz + zx + xy=S3y {y + «)(» + x){x + y) = 294. 33. («-2)»-|.(y-3)2 + («-l)2 = 2'4, xy 4-vz + zx = 63, / ^x + Sy + z = 30. 30, 12. / I- j\XA/\^ 481 481 = 10, / = 33, / = 294. ■y\ = 3./ = 7, CHAPTER XIV. ELIMINATION. 209. From two or more equations it is frequently necessary to form an equation in which o.e or more of the quantities pre- viously involved will not appear. The process of finding such an equation is called Elimination, and the unknown quantity or quantities which disappear are said to be Eliminated. 210. In order that elimination may be efiecfced there must be at lea^t on. tndependeni equation more than the quantities to be •f ciindrtitn '"'"^''''^ ''^''''^'°'' '' °^*''' "^^^^^ ^"^ Equation Thus from and a^x + b^ = Cj (1) (2) we can obtain the values of .. and y, and these values, substi- tuted m ' will give an equation without . and ,, but expressed in terms of the other letters of the equations. This new equation expresses (2) a^tsT" ' ^ """^ ""^''^^ *^' **""'' '^"^''°"'' ^^^' 211 It often happens that the student does not at once per- ceive the number of quantities to bo «Umi-„af.^ ,•„ *u v/ given him. Thus, in the following problem : 204 HIGHER ALGEBRA. Eliminate Xy y and z from (1) (2) (3) the beginner usually assumes that there are three independent quantities to be eliminated; and as there are only three equations given, he naturally concludes that elimination is impossible. In such problems, however, there are only two independent quanti- ties. For, divide (1), (2) and (3) separately by z, then we obtain, \ z z CC 11/ «i • - + />i • - + cj = 0, z z a . --i-b . ^ + cj = 0. z z (4) (5) (6) Thus we see that in (4), (5) and (6) there are only two inde- pendent quantities, namely, - and -, and elimination is possible, z z If, however, (1), (2) and (3) had, on the right-hand side of the sign of equality, quantities such as d, d^, d^, elimination would be impossible, as there would now be three independent quantities to be eliminated, and only three equations. 212. No general rule or rules can be given for Elimination. In simple cases the values of the quantities to be eliminated can be obtained from the same number of equations, and then these values can be substituted in the remaining equation or equations. This process, however, is not alVays easy or desirable, and various artifices are employed to get rid of the undesired quantities. The following examples may give some assistance : (1^ (2) (3) (4) (5) (6) ELIMINATION. Ex. i.— Eliminate x from the equations, ax + b = c From (1), ^-^ and from (2), x = - X- a Ci~bi a, ,, which is one form of the desired result. Ex. 2.—li and eliminate x and y. From (1), and from (2) ax-\-by = aiX + bji/ = 0, a I a a, x' x' ^ 205 (1) (2) 0) (2) = -~y or aftj - aj) = 0. b bi Ex. S.~Ii and ax + bi/ + cz = (1) (2) eliminate z, and find the value of - y Multiply (1) by q.and (2) by c, and subtract the products. Then x(aci ~ «ic) + y(bci - b^c) = 0. Therefore i 206 Ex. 4- — Given Prom (1), from (2), and from (3), HIGHER ALGEBRA. x + y + Z'^a, (1)' xi/ + 2/z + zx=:b, (2) xyz=^c, (3) J y + z = a-x, b-yz X e yz=-. X eliminate y and z. Combining these results we obtain .r^ - ax"^ + bx -c = 0. Note.— The student's attention is directed to another and more elegant solution of this problem in the chapter on Simultaneous Equations. Ex. 5. — Having given a? = fty + c« + c?w, y = aar + c« + c?w, » = aar + Jy + c?w, w = aa; + Jy + c«, a b (1) (2) (3) (4) show that 1 = + d a?, y, 2, u being supposed all unequal. Adding ax to both sides of (1), by to both sides of (2), cz to (3), and du to (4), we obtain a:(l + a) = aar + iy + c« + c?w = 3/(1 + J) = «(1 + c) = m(1 + (/). Let axJfby-\-cz-\-du = h. h k Then x = - y- z- 1+a' ^ 1+i' -"l+c Substituting the values of a;, y, «, w in a;(l •\-a\ = ax-\-by-\-cz + du, ak bk ck k . k , and u = \+d' we obtain k = Dividing by A;, 1 = dk \+a^\+b'^\+c\+d' d \+a^\+b'^\+c\+d' II <1) (2) (3) (4) • " BLIMINATIOK. Ex. 6. — Find the relation between a, A, c, having given sa7 that is, « y X 7i ' y X «» y' a:' «2 y» 22 = aic + 4. or a'' + 62 + c8-a6<5 = 4. Ex. 7. — Eliminate x £rom the equations, a^ar^ + h^x + Cj = 0, As in Art. 39, Exs. 1 and 2, we have x" X 1 icj - ijC caj -Cia ab^- a^h' :. 0^= bci — biC and a: = ca, — c,a or abi-Uib abi -a^b' , / cai-Cia y_ bCi-bjC \abi-aibj ~ ab^ - aji' (ail - a^by{bci - he) = (ca^ - Cja)'. (1) (2) / II 208 mOHER ALOEBRA. Ex, 8. — Eliminate a?, y, % from (ar - y + s)(y - 2 + ar) = oy«, {y-z + x){z-x-^y) = hzx^ (« - ar + y){x -y + z) = cxy. Multiplying together (1), (2) and (3), {(ar - y + z){y -z + x){z -x + y)}« = ahc^y^z\ or or or cr or 0) (2) (3) {x-y-{-z){y~i.^x){z-x^-y) _ ^— xyz ^' x'^y + a?z + y h + y^x + z'^x + z'^y-x^-'}r^-7?-2xvz ,-r- ' — . — •_ _ Vaic, xyz ' ixyz-x{:^~(y^zy}-y{y'-(z-xr\-z{z'-(x-yn ^ ^— xyz ' 4:Xyz - axyz - bxyz - cxyz ,—r— ~ ■ '— — V abc, xyz i — a — b-c= Vabc. /. (4:-a-b-cy = abc. \ EXERCISE XX. e.1. Eliminate x and y from x + y = z, x^ + i^ = a^, aP + y^=b^. ^ 2. Elimiuiite x and y from x + y = a, xy = z\ a?^ + y' = h\ ^ 3. EL^ii^ate x, y and z from the equations, a^(y + «) = a3, y'(a; + «) = i3^ 2'(a; + y) = c', xyz = abc. ^ 4. Eliminate »», w, />, q from the equations, ^-y y + y 'pm qn jir? n^ P^ 9' ■. / ,1.6. Eliminate x and y from the equations, aa? + iy = ar + y + a;2/ = a:2^y2_1^0. / II ELIMINATION. 209 0) (2) (3) be. abc, h^. i J: 6. Given y^-x*=zay- hx, 4xy - ax + by and .r' + y* = 1 , elimi- / nate x and y, and show that (a + i)8 + (a - A)* - 2. 7. Eliminate x and y from the equations, / x^+y^^a, b^x + Sx^y^, c = y + 3x*y*. C3. Find the condition that / «iar + % = Ci, a^ + b^ = c^ and a^ + h^^c, may be satisfied by the same values o! ar and y. 9. Eliminate x, y and » from tho equations, (b + c)x + (c + a)y + {a + b)z=>>0, (i -c)ar + (c-«)y + (a- % = and a:-^ + y-i + 2;-i«o. 10. Eliminate a, y, s from aar + iy + c« = 6a: + cy + a2 = car + ay + i»=l when ar' + y2 + «2=p2. 11. Eliminate x, y, « from ha' + ey + az^cx + ay + bz'=ax + by + cz = ab + bc + ca and ar + y + 2; = a + 6 + c. 12. If ax + cjy + biz = 0, CiX + by + a^z^O, b^x + Uiy + cz = 0, show that aa^^ + bb^ + cc^ = aic + 2aiUiCi and {ab - c,^{ca - bi')x' = (be - a,^){ab - c,'),/ = (be - a,')(ae - b,y\ 13. Eliminate m, x, y, « from the equations, y + z + u==ax, z + u + x=by, u + x + y = cz, x + y + z = du. 14. Eliminate a and b from the equations, a- y=« « . S' A k ;r2+A2=i' ;; + i = i. - + 7 = 1. a a a 15. Eliminate x or 77 from t.bfi K^nnnfinno I 210 HIGHER ALGEBRA. 16. Eliminate ar or ?/ from {x-y){x''-,f)^^ and {x + y){x' + y^) = %^, 17. Eliminate x or y from a^ + 2/' + (2y/2+l)(a; + l) = and x" -f -xy -\=0. 18. Eliminate x and y from a.x + hy = c\ xy^,?-ah and -, + 2^= 1 a-* W>3 g • 19. Eliminate a: and y from a.r2 + />a:2/ + c2/2 = and «i.c^ + />,j:7/ + qy^ ^ 0. 20. Eliminate or and y from •^' + ^2/ + y = «^ x^ + x'f + y^^h'' and .r« + a^y + / = ^s. 21. Eliminate ar and y from «' 2^_. a^ y' , a:* 2/* m a n 22. Eliminate a? and y from a%y = mb\ -2 + |2=l and ht + my=p. 23. Eliminate ar and y from r(iar - ay) = .^^(aar - hy) = ary and a^ + / = c\ 24. Eliminate .r and y from « 1 "* o . "^ n . ex by x + — = 2tn, y + ~. = 2n and —+-£ = *, X y y ^ r 25. Eliminate x and y from «.r + 6y = c(a2 + 2/2)i and a^x^\y ^clx" + y'')^. 26. If 2/2_wi(2ar + m) = a2, x^ - m{2y ^ m) = h'^ and « + a? = i + show that m = or m = a + i. y, ELIMINATION. 211 4- w. 27. Find the equation between a?, y and a, independent of a and 5, from the equations, a? + ax = y^ b^ + bx = z and a'^ + b^=l. 28. From aar + iy -« = (mV + 712^2 + c*)i = n%-i = m^aa?-i find an equation which does not contain a and b. 29. Given the equations, x^ = -^— and y = = 5, show that if 1 + i5 1 — 2; 2; be eliminated the following equation will hold between x and y : {^ + {f + l)^^ + {y-{f + l)^}^=x-^-x. 30. Eliminate y and z from the equations, y + « = o, a;2 + y2_2ma?y = 62 a,nd ar^ + 2^ + 2/»a:« == c^, and show that (« + ^ + <^)(^ + ^-«)(<^ + «-^)(« + ^-c) .^^^, 4a2(l - m2) 31. Eliminate a from the equations, — — -„ = -~— = , and prove that x{y'^ - ^2) + 2y{z'^ -a^) + iz{x^ - y^) = 0. 32. If ax^ + hf=bz{y-x) and bx^ + ay^ = ay{x ^ z), find the relation between x, y and a. 33. Eliminate a;, y, and z from ar + y + « = a, a:^ + y2 + 3;2 = ^2 ^jid a;^ + yj + ;53 _ 3^^^ ^ ^s^ 34. Eliminate ar, y and » from ar + 2/ + a = a, x"^ + y"^ + z"^ = b\ u^ + y' + z^'^c^ and xyz = d\ 35. Eliminate ar, y and » from a: + 2/ + « = and -+-=- + £ = _+_. a X y b z c 36. Eliminate ar, y and z from ar2/2; = a«, (;r + yXy + a)(a + ar) = c», -+^ + -=aand-+2^ + -=6. y z X z X y 37. Eliminate a*, y and « from x^^y^-^rz^- layz, y"^ = .r2 + 2;2 - 262aJ and »2 ^ ^2 ^ ^2 _ 2cary. / 212 HIGHER ALGEBRA. 38. Eliminate x, y and z from 39. Eliminate ar, y and z from (ar + y)2 = 4c:ry, (.r + ^)2 = 4i:r« and {y + zf^^ayz. 40. Eliminate :t, y and z from ^^-y« = «, y^'-^.r^i, z^-^cy^c, ax + hy + cz = d. 41. Eliminate .r, y and « from ^'^ y^ Z^ X^ y"^ 2 42. If X = a. y+ z ■ z + x tween o, i and c. y ^ = A and =c, find the relation be- x + y 43. Find the relation between a, h and c, having given 44. Eliminate a and i from the equations, a'-.^ 2ar + 3y ^^^^^r^y' <''-^' = {^-yy and «Ui^=^*. 45. Eliminate x and y from x + y==a,c^ + f=.h' and ;r« + y= = c«. 46. Eliminate x from the equations, 47. Eliminate .r, y and z from the equations, ^ , y z X - + - + -=«, - + y a X ' z ^r*•(^f)(f-^)(^i)=«. 48. Eliminate a; and y from 4(ar=' + y«) = aa. + Jy, 2(0:^ - y^) = ax - iy and xy = c^. «i = abc. VI o'^z. he- = 0. = C6. c. CHAPTER XV. THEORY OF QUADRATICS, 213. In Part I., Chapter XXI., the simpler portion of the Theory of Quadratics has been ti-eated. It remains now to give that part of the Theory which is of a more difficult nature. 214. In algebraical researches it is frequently necessary to determine for what values of the variable a particular quadratic expression becomes positive or negative. A few numerical illus- trations will render the principal proposition more easily intelli- gible. Take for example the expression, a-^ - 13a; -h 36, and for z substi- tute various numbers, anc! collect the result. Thus when x= 1, 2, 3, 4. 5, 6, 7, 8, 9, 10, 11.... :r2-13ar + 36 = 24, U, 6, 0, -4, -6, -6, -4, 0, 6, 14.... It will be observed that when a; is 4 or 9 the expression vanishes, because these are the roots of the equation, a;2_ 13x4-36 = 0; that when x is between those roots the expression is negative, and that for all other values it is positive. If we change the sign of every term in the expression, the vanishing points are unchanged; but the expression is now positive when x is between 4 and 9, and negative for all other values. Next consider the expression, 2x^ ~8x + 25. It will be found that this expression is always positive for real values of x. When a: = 2 the expression becomes 17; but for all other real values of 214 HIGHER ALGEBRA. X it is greater. The reason for this peculiarity is seen at once by writing the expression in this form: ^{x-2Y+^]. When x=2,{x- 2y is zero; but for all other real values of x it 18 a positive quantity. Note that in this case the roots of are imaginary. We will now proceed to state and prove the general proposi- tion relating to this subject. 215. The quadratic expression, ax' + bx + c, is always of the ^ same sirjn as a for all real values of x, except wJien the roots of the quadratic equation, aaP + bx + c = 0, are real arA UNEQUA^'ami X lies between them. * ^^ (^x'^ + bx + c = a{x-m){x-n). Then the roots of ax^ + hx + c^O &rem and n. (Art. 294, Pt. I.) 1 Let m = M. Then ax'^ + hx + c^ a{x - nf. But {x - nf is posi ive when x and n are real; .-. a{x-ny is positive if a is positive, and negative if a is negative. 2. Let m>n. Then if x>m and >w, {x - m){x - n) is posi- tive, since both factors are positive. Also, iixm and n, then {x-m){x-n) is nega- tive, since one factoi is positive and the other negative. When {x-m){x-n) is positive, a{x-m){x~n\ as shown before, is of the same sign as a; but when {x-m){x-n) is negative, then a{x - m){x - n) must have a different sign from a. 3. Let the roots oiax^ + hx + c = be imaginary. Solving ax^ + hx + c = Q we find 2a^ b \/W - 4ac 2a If i^<4ac, the roots must be imaginary. THEORY OF QUADRATICS. 215 \ Now, ax'+bx + cm^a(x'+-x+S.\ \ a aj But (.+ i-)'is positive; and '-^ i. positive, since «V4ac and M U positive. .-. ar- + 6^ + . equals the product of « and a posi .ve quantity ;.-. „;^ + 4^ + „ „„,t b^ „f th^ ^^^ when the roots of ax' + hx + c are imaginary. Ex. i._The expression, 2.r' + 8.. + 9, is always positive for all real values of x. Since 8'c 9, the roots of 2:r= + 8^+9 = are imaginary; the value of 2^ + 8.+ 9 for all real values of . is of thTcaie Sign as 2, and is therefore positive. ..n?; ^'~?^^ JW^^-ion, ix^+Ux+U, is always positive ex- cept for values of x lying between the roots of 4^^ + 1 5^ + 1 2 = Forl5>4><4xl2; .-. the roots of 4^+ 15.+ 12 are real and unequal, and, therefore, ix^+Ux+U is negative when x lies between these roots. ■^1 .216. It is sometimes possible to find maximum and minimum ^ Values of a fraction for real values of the unknown quantity. Let it be required to find the maximum and minimum values of Assume 2ar»+2ar+r x'^-Zx-Z = y&. 2a;2 + 2a?+l Multiplying out, and arranging as a quadratic, (1) 216 HIGHER ALGEBRA. Since x is real, (3 + 2ky + 4(1- 2k){Z + yfc) ^ 0, 21-8A;-4F^0, or or or 4P + 8yfc-21<0, (2A + 7)(2X;_3)^0. Now in order to be negative or < 0, 2k must be < 3, and .*. A;< - . so that the maximum value of yfc is -. Also, if 2A;<3 2^ + 7 must be positive; .-. 2A> - 7, and :.k>J-. The mini- mum value of k is therefore - | , and the maacimum value, ?. If J^=--^or ~,the expression, (2A; + 7)(2^ - 3) = 0, which is the condition that the roots of (1) should be real and equal. '' ■j fc ^^ '- ? ^^' ^^ -^^^ '^^ condition that a^ + 2hxy + h^f + 2gx+^y + c .ji^Tyrmy resolve into two rational factors, the constant quantities, I «j ^> o, gyj, c, Jeiri^ rational. ^ The expression, a.:^ ^ ^j^^y ^ j^, ^ ^^^ ^ ^^^ ^ ^^ ^.^^ ^^^^^^^ into two rational factors if the equation, «^' + 2Aary + 6y2 + 2^a?+2/y + c = 0, (1) when arranged as a quadratic in x or y, has rational roots. Arrange (1) as follows: «ar2 + ;r(2% + 25') + (i/ + 2/y + c) = 0. When the equation is solved tlie quantity under the radical sign will be {^hy + 2gY-ia{hf + 2fy + c). - || In order that the roots may be rational (2^ must he a perf«qt square. ' ^ -*-^ t» B < 3, and o, if 2A;<3, The mini- ilue, ~. If h'.ch is the il. jx+^y + c quantities^ ill resolve (1) )OtS, le radical B. perf«Qt THEORY OF QUADKATICS. Now, (2) = 4 {f{h? - ab) + y{2gh - 2af) + (j^' - ac)}, the condition of which being a perfect square is (2gh - 2a/)» = i{h? - «*)(/ - ac\ 217 or or 218. To find the condition that a:t' + bx + c = nmy have roots equal numerically, but of diflferent signs. Since then a3p + bx + c = 0, -2 . * c ^ ir+-x+-=0. a a The sum of the roots of this equation is - - . a* But since the roots are equal numerically, and of different signs, the sum of roots = 0. b " "a'" ^^ ^'"^ ^^*''^<^o^ u t/6. If n be real, prove ^~~ must lie between 3 and I / 7. The greatest value which ^^) admits of for any real / 4/9,7 .ti-« •' valueofaris— ^— . ^ 8. Show that for real values of x, ^ ~ ^^ '*' ^ tween and - 4. - 4a? + 4 ,, ■^^T cannot lie be- ^ i^. Withi] t^. Within what limits is the value of ^~-^ an integer or / a rational fraction ? t/10. Show that {x - aXb - a,) can never exceed -(a - bf. / 11. Show that the least value of fc±^)fe+l) j^ ^J ^ ^^^^ ^^^ the greatest value of ^i±^fc±) is ^ + *)' iab • X' 12. Show that the roots of (3a -x)-^ + (36 - x)-' + (3c _ x)-^ = are real if a, b, c are real. 13. Show that by giving an appropriate real value to x, ^ 4^^-36^4-9 12ar2-f-8xTl can be made to assume any real value. 14. The expression |^^|^|, admits of all possible values, / provided that one, and only one, of the quantities a or b lies be- tween c and d, and otherwise will have two limits between which it cannot lie. / real value / 3 and -. )r any rea/ ^ not lie be- ^f integer or ^ h)\ / + h^)\ and / •le values, ,/ h lies be- jen which THEORY OF (QUADRATICS. 219 15. The expression, ^^^^^^^ will be capable of all values for real values of a-, provided that («c, - «,o)'< 4(a,6 - ai>,){h,c - hc^ 16 If a be >\ and c be positive, prove that the greatest value which the expression, (x - «)(:. - A)(x - a - c)(a: - 6 + c), can have tor values of x between a and h is 16 17. Find the limits between which a must lie in order that aar^ - 7ar + 5 5x* - 7a; + a may be capable of all values, a; being any real quantity. ^8. Show that |— ^ will be capable of all values when ^ X is real, provided that p has any value between 1 and 7. 19. If the roots of a^^Ux^c^^ be possible and different then the roots of (a + c)(ar2+ 26a: + c) = 2(ac-6WH- 1) will b^ '" K impossible, and vice versd. 20. Show that ff,l2al^^Ia) ^"^ ^^ ^^P^^^^ «^ ^" ^^l»e« ^i^en X is real, if (c^ - c?') and (a^ - b') have the same sign. C21. For what values of m will the expression, ^ f+2xy + 2x + my~3, be capable of resolution into two rational factors? j 22. Find the values of m which will make ^ 2x^ + mxij + 3/ _ 5y _ 2 equivalent to the product of two linear factors. /• 2.3. Show tlia>. »vi/^ ^,2\ ~,.// --\ 1 J •. a . ,s- ;" ^"V" -:/ /- -^uv - -V itivvays admits of two real linear factors, n I ! ¥ 220 HIGHER ALGEBRA. 24. Find the condition that ' Ix^ + mxy + nf and lix"" + m^xy + n^i/'' may have a common linear factor. / 25. If the expression, 3.r'- + 2Axy+ 2?/ + 2ax - 4y + 1 = 0, can J be resolved into linear factors, prove that A must be one of the roots of the equation, A"^ + iaA + 2«' + 6 = 0. ^26. Find the condition that the expressions, ' ai^ + Uxy + hy^ and «..r» + 2h^xy + by, ^ may be respectively divisible by factors of the form y-mx and my + .r. . 4. 27. Show that in the equation, ar2-3ary + 2/_2ar-3y-35 = 0, / for every real value of x there is a real value of y, and for every real value of y there is a real value of x. 28. If X and y are two real quantities connected by the equa- tion, ^a? + 2xy + y - ^2x - 20y + 244 = 0, then will x lie between 3 and 6, and y between 1 and 10. ^ 29. If (ax' + bx + c)y + a,x^ + b,x + c^ = 0, find the condition that X may be a rational function of y. 3d. The expression, B ax^+bx + c ^ f T ,"Vill be capable of all values cx^ + bx + a whatever if b^>{a + cf. There will be two values between which it cannot Jie if b'^<{a + cy and >4ac, and two values between which it must lie if 62<;4a(.. 31. Find the greatest numerical value, without regard to sign, which the expression, {x-S){x - 14)(x- \Q){x - 22), can have for values of x between 8 and 22. L / THEORY OP QUADRATlCa 221 1=0, can J one of the - mx and for every the equa- 5 between ition that ill values ;en which between 1 to sign, have for / SOME RELATIONS BETWEEN ROOTS AND THEIR COEFFICIENTS. ^^19. Theorem.— 7/- the coefficients of an equation are rational khsn surd roots must be in pairs; i.e., ifa+Vbis one root, then a- V b must be anotJtsr root. Ex.—l^ta+ V~bhQ a root of a^+px' + qx + r^^O. Then, substituting a + V~b iov x we obtain (a + VTf +p{a + VTf + y(a + ^Z 1) + ^ ^ q, or {«» + 3a* +pa' +pb + qa + r} + {Sa^ + h + 2pa + q]VJ=. 0. But when the sum of a rational quantity and a surd is equal to zero, the rational quantity must be equal to zero; also the surd equal to zero. .'. a^-¥Zab+pa'^+pb + qa + r = 0^ n) ^^ {^a'' + b + 2pa + q)VT=.0, (2) Subtracting (2) from (1), a? - 3a» x/J+ Sab - b VJ+p(a^ - 2a VJ+ b) + q(a - i/J) + ^ = 0, '*'' (« - "^'^Y +P{a - ^If + q{a - i/ 6) + ^ = 0. From this it is evident that (a - \^T) is a root of The following is the General Froof.~Let the equation, x''+px"-' + qx«-^ + .,,, _^ root, u+ Vb, and let the coefficients, have rational. Py 9, be 'ir 4 II ! 222 HIGHER ALOEnnA. If a + 1/ lis substituted for x the equation will take the form of P+Q VJ=0, where i* stands for the rational terms and Q VJ the irrational. Since p+Q^J^o^ :. P = and Q\/'b = 0. :. p-qVJ^o. But a a- V J he substituted for x in the equation the result obtained will be P-Q\^b; and as P-QVI^q, a- V b nmst be a root of the equation. See Art. 186 for a shorter proof. ^NL^-^20.— Theorem.— 7/* the coefficients o/an equation are real, ■^J\lhen ifa^h^-1 is a root, a-bV -lis also a root; or, in brief, if the coefficients of an equation are real, imaginary roots occur in pairs. • The proof of this proposition can be obtained by the same line of reasoning as in the preceding theorem. ^a;.— Let a + J V - 1 be a root of a^ + joar + ^ = 0. Then (« + *VTi)2 + ^(a + 5^-n)^.^=,0, ®^ (<^''-^^+pa + q) + {1ab+ph)V~^ = 0, °'' a'-b'^+pa + q^O ^"^ {2ab+ph)V~:^=.0. Subtracting, «" _ ^ab /Tl „ y" +p{a - b V^) + y = 0, {a-bV -\f+p{a-bV ~\) + q = 0. From this condition it is evident that a - 6 v^ - 1 is n. rnnf of a^-\-px + q = 0. the fonn of } irrational. 1 the result - Vi must proof. n are real, oTj in briefs ots occur in B same line > a root nf THEORY OP QUADRATICS. 223 HXBROISE XXII. Solve the equations: 1. 3;r« - 10^+ 4.r» - :r - 6 = when one root is i±-^li ^ 2 2. 6a^-13r'-35ar2- r + 3 = when one root is 2- V^. Q^ a^+ 4«» + 5ar- + 2ar - 2 = when on« root is - 1 + ^rj. ^ 4. x«-^*4.8x'-flx-15 = 0, o^e ^t l^eii^g V 3 wid a*,^er ^ 5. Form the equation of the lowest dimensions with raUoual coefficients, one of whose roots is: "' C. (1) V^S+v^T^; c (2) _ i/^+-v/5; (3)-i/2-V"rT; (5,(4) ^s-f^ve. ^ 6. Form the equation whose roots are ±4 V'S, 5±2 V^". V 7. Form the equation of the eighth degree with rational co- efficients, one of whose roots: is V 2 + V' 3 + ^1"!. 0^8. If a±|3l/-l betherootsof ar» + 5^a: + r = 0,then|3' = 3a'+,j. 9. If r»+;?ar2 + ja: + r = be satisfied by a: = 3+ VT, it will bo satisfied by a: = - r. 10. If - + '^'hh^B.xootoi^^p^^qxArr^^.t^^n a^->,j>a4,fi is a factor of r, i not being a perfect square, and ;>, y, r rational. 11. If a + 6A/TT be a root of y + yar + r = 0, th^ a is a root of 8.1:3^2?^- r = and Za^-h^^ -^, but it « + 6 1/31 boa.root of ar» - ^^2 - r = 0, then a is a root of 8x^ - Spx" + l^x +r = D. 12. If a:» + g'a: + r = have a root \(xi^ V 6), ghow that «is a root of the equation, ar* + ga: - r = 0. CHAPTER XVI. i! I INDETERMINATE COEFFICIENTS. 221. The student of Algebra is frequently called upon to de- termine the relations between the roots of an equation and its coefficients; also the conditions that must be satisfied by the co- efficients of an algebraic expression when it vanishes for certain definite values of the unknown quantity involved. Bx. — What condition must be satisfied by the coefficients of aa:' + f>x + c if this expression vanishes for more than two given values of x ax^ + bx + c l>eing a positive integral function oi x1 Let ax^ + bx + c vanish when x = m, nfi. Then a(^x^+ ~x+^j =a(x-m)(x-n){x-py, Part I., Art. 275 therefore ^ V ^ a ^^ a ) "^ ^^ ~ "^^(^^* + n+p) + ax{mn + np +pm) - amnp, a quantity of the third degree, which is impossible unless aar' = or a = 0. If a = 0, then ax^ + bx + c = bx + c. In the siime way it can be shown that 5 = 0, and .'. c = 0. The conditions, then, for ar' + L. + c vanishing for more than two Values of x are a = 0, 6 = 0, c = 0. £.<■£.. TT c p-uj;.-ccu nuw tu give i/iiu proor or vne general proposi- tion of which the preceding example is an illustration : INDETERMmATE COEFPICIEMTS. 225 NTS. upon to de- >tion and its sd by the co- 3 for certain oefficients of n two given jtion of ar ? I., Art. 275 nn) - amnjj, nless '. c = 0. r more than 5rai proposi- Proposition. — If any positive integral function of x of the 7i'* degree vanish for more than n different values of x, then each of the coefficients must vanish. Let ^„a;" + ^„_ia;"-^ + . . . ^lo be a positive integral function of x. This function can be denoted by the symbol yi[a-). Let «!, cfa a„ be values of x which make /(a-) vanish. Then f{x) = A„{x - a^){x - a^{x - ag) . . . . (x - a„). If possible lety^a:) vanish for x = a„^i. Then y(a„+i) = = il„(a„+i - ai)(a„+j - a^) . . . . (a„+i - a„). Now, since or„+i-Oi, a^^^-a^ «n+i-«n are each not zero, since fl„+i is different from a^, a.^ a„, /. A„ must = 0. Therefore /(a?) reduces to A^_.^x^-^ + . , . . A^. In the same manner .<4„_,, ^„_2. . . . Aq can be shown to be zero. The above is the usual proof; but it is well for the student to recognize that the truth of the proposition depends upon the fact that a positive integral function of x cannot have more linear factors than is indicated by the highest power of x. If it has, then the coefficients of the function must be separately zero. 223. Corollary.— //"id + Bx + Cx'-\-... .Nx^=^A, + B^x + C^^ + ifiic" for more than n values of x, tJien A=Ai, B = Bi, For, transposing, {A~ A,) + {B - B,)x + . . . .{2^- Ni)x'' = 0, and the equation is satisfied by more than n different values of x. Therefore the coefficients separately vanish; i.e., A-A, = 0, B-B, = O....N-N-^ = Q^ A=A„ B = B„ C=C,.... JV=F,. Ex. 1. —Let x^-p.v'^ + qx-r=^{x-2){x- 3)(a; - 4) for more than two values of x. Then, since {x - 2)(ar - 3)(.c - 4) = x^ - ^x^ + 26a; - 24, we find, by equating coefficients, that ;? = 9, y = 26, r = 24. I i__i_:. ' 226 HIGHER ALGEBRA. Ex. ^.— Let ax*+p3r^ + qx'^ + rx + 8 = 4:u!^ + 3z^ + 5x-2 for more than four values of x. Then a = 4, j(j = 0, 5 = 3, r=5, s=-2. This principle is called that of Indeterminate Coefficients, by which is meant '• coefficients that have to be determined." As the principle has a very extensive application we append a few more solved examples. Bx. 3. — Show that the following is an identity : a\x -h){x-c) Ir (x - c)(x - a) c\x - a)(x - b) {a-b){a-c) {b-c){b-a) "*" (c - aXTTftjT " ^• From inspection we see that the equality holds when for x we substitute a, h or c. But the expression is of the second degree in x; and since it is satisfied by more than two difierent values of x, the coefficients of corresponding powers of x on opposite sides of the equation must be equal. Therefore the equation is an identity. ^ Ex. 4' — Find values for a and b which render the fraction, 2x^+(a-b)x + 2a2 - U^ 3^2 + (a-7)x + 3{a^+ 2ab + Sb^ the same for all values of x. Let 2.t' + («- ').r+2o2_3i2 Sx^ + {a-7)x + 3 (a" + 2ab + W) where h is the same for all values of x. = k Clearing of fractions, and arranging as a quadratic in x, we obtain x'{2 -U) + x{a-~b- k{a -1)]+2a'- W - Zk{a'' + 2ab + W) = 0. Now, since the left-hand expression vanishes for all values of x, the coefficients separately vanish. 5a; -2 for more Coefficients, terniined." As 3 append a few 1-b) = x'. -b) when for x we ( second degree lifferent values X on opposite the equation is he fraction, in X, we obtain all values of x, Indeterminate coeppicients. 227 Therefore (l) 2-3k = 0, (2) a - 6 - k{a - 7) = 0, (3) 2a2-362_3>fc(a2+2a6 + 362) = 0. 2 From these equations we find that k=-, a= -6 or -14, 6 = 2f or 0. Bx. 5.— Resolve 2x^ - 2\xy - 1 \xy^ - ar + 34y - 3 into rational factors of the first degree. Assume Ix'-nxy-Uy^-x + Z^^y-Z^ {2x + my + n){x + ry + s). Multiplying out, and equating coefficients of like terms, we have -2l = 2r + m, — 11= mr, -l=w + 2, 34 = nv 4 ms, - 3 = rw, (1) (2) (3) (4) (5) Solving these equations we find m=l, n- -3, r= -11, and « = 1. .'. factors are (2x + y- 3)(x - 1 ly + 1). This is a tedious process of obtaining the factors of this ex- pression, but it is here introduced to illustrate a method some- times useful. Ex. 6. — If a, 6, c, d are the roots of x^-px^ + qx^-rx + 8 = 0, find the relations between the roots and the coefficients. If a, 6, c, d are values of r that make x^ - pi^ + qx"^ - rx ■{■ s vanish, then x-a, x-b, x-c and x-d are factors of this ex- pression, and therefore a^ -px^ + qx'^-rx + 8 = {x- a){x - b)(x - c){x - d) for all values of x. 22S HIGHER ALGEBRA. Multiplying out, and equating coefficients, we obtain a + b + c + d = j», ab + bc + cd + da + bd + ca = q, abc + bed + cda + adb = r, abed — s. (1) (2) (3) (4) (1)» (2), (3) and (4) are the relations required. From this ex- ample it is easily seen what general relations exist between the roots and coefficients of positive integral equations. Hx. 7.— If x" B C -, + {x - a){x - b){x - c)~ X - a x-~b x-c^ find the values of -4, B and C, and prove that abc (1) = 0. (a - b){a-c) {b -a){b-c) {c- a){e - b) Clearing (1) of fractions we obtain a^ = A{x- b){x -c)+ B{x -a){x-c) + C{x -a){x- b). (2) Since (2) is an identity the equality holds for all values of x. (3) Therefore when x = a, a^ = A{a-b){a-c), or A = Also when x = b, P = B{b-a){b-c), or B = Similarly when x = c, e^ =C{e-a){e-b), or C a' {a- -b){a- -«) (b- .a){b- c2 -«) (c -a)(c~ b) • (4) • (5) ABC Again, in (1) let a: = 0; then — + — + — = 0. abc But from (3), (4) and (5), by dividing (3) by a, (4) by b and (5) by c, we find that a b c~{a-b){b-e)'^{b-a){b-e)'^{e- a){e - b) = 0. 0) (2) (3) (4) 'rom this ex- between the (1) = 0. :-b). (2) values of x. • (3) • (4) • (5) !(«- -«) h' ){b- -«) ,2 (c-b) y 6 and (5) = 0. -h) INDETERMINATE COEFFICIENTS. 229 224. The principle of Indeterminate Coefficients is often em- ployed in finding the sum of a series. Ex. ^.— Sum the series, p + 22 + 32 + 4^ + . . . . n\ Assume V+2'' + 3' + 4:' + ...n^ = Ao + A,n + A,n'' + A,n? + A^n' + ... (1) Then, since the sum of the squares of the first n natural numbers is a function of n, V + 2^+Z^ + i^ + ...n' + {n+iy = Ao-rA,(n + l) + A^{n+iy + A3{n+lf + Ai{n+iy + ... (2) Subtracting (1) from (2), {n + iy=Ai+A2{2n+l) + Ar^{3ri'+Sn+l) + A^(irv^ + 6n'' + 4n + l) + ... or n^ + 2n+l=Ai + Apn+l) + As(3n' + 3n+l) + (3) It is not necessary to write down any coefficients beyond A^, because J 4, A^, A^.. . are the coefficients of w^ w* . . . in (3), which do not exist on the left-hand side of this identity, and .*. must be =0. Equating coefficients of the same powers of n, l=Ai + A^ + As (term without w), 2 = 2^2 + 3^3 (coefficient of w), 1 = 3,43 (coefficient of n^). Solving (1), (2) and (3) we obtain A -^ A ^ A ^ ^3-3, ^3=2' ^1=6- -A (1) (2) (3) 1 ... V+2^ + 3^ + i^ + n\.. = A,+ ~n+l-n'+^n\ o 2 II i- (if 230 HIGHER ALGEBRA. Since this is true for all values of 7i, let n = 0. :. P+ 22+32 + 4=^ + . 9 n w^ j^3 6^2^3 w + 3^2 + 2n' = ^ (1 +3w + 2w2) 6 _ n(l+w)(l +2n) 6 • EXERCISE XXIII. 1. If «!, «„ a3 be the roots of ^+px'- + qx + r-^0, express, in terms of p, q and r, <^^^^ ^+i;+? ^^> «x^+«/+< (3) ^^+^^+^2^«2^i3 % ^ ^ «2 as «3 «i «! tta C 2. If «, i, c, ^ be the roots of ,^-r^+x+\ = 0, find the values of (1) a'h + a:'c + aH+b''c + .... (2) a^ + b' + c' + d^ 3. If o, b, c are the roots of or^ +^,.^2 + g^ + ^ = 0, form the cqua- '^ tions whose roots are (1) a\ b\ of INDETERMINATE COEFFICIENTS. 281 8. If a, J. c are the roots of ^ +;,a:=' + ^.r + r = 0, form the equa- oions whose roots are (1) h + c,c + a,a + b- (2) bh\ c^a\ a^b\ 9 The roots oi ^ + q^ + r=.0 ^ve denoted by «, 6, c; form the equation whose roots are ha + ac, ch + ba, ac + cb. that the equation, ^ + ^^ + ^ = 0, may be put in the form ^^{x' + ax-^.by, and hence solve the equation, 8^ - 36.r + 27 = 0. 11. Investigate the relation which exists between m and n when ^^-{2m^^,ny^im^^,mn).-,m^n is a perfect cube 12. Determine the relations which exist among a,b c d e v a when a.^ + b^^c.^^d. + e is divisible by .^+ A; ' ' ' ''^' ^ 13. Investigate the condition for the expression, 4.r* - ip^ + 4^a;2 + 2p{m + l)x + {m+ l)\ being p. perfect square. 14. Investigate the condition for the expression, Aa^ + 2Bxy + Gf+ 2Dx + 2Ey + i^, being divisible by a factor of the form ax + by + c. 15. Express i{x^ + a- + ^ + x+\) s^ the difference between two squares. 16 Investigate the relation between the coefficients that the equation, a^^bx^+cx+d^O, may be put under one of the forms, (1) x' = {x''+px^.q)\ (2) q'^ix'+px-^qf. Solve in this way ^a^-x"^ -2x+l=0. 17. If two of thfi rortfo n( r,^ I Qi^a , o , t -. -" "■ ""^ T ^^j>- \ OCX ■\-a = \i are pniml prove Kac-h'^){bd-<^)^(<^d-hc)\ ^ ' 11 fr Hi J 232 HIGHER ALQEJRA. 18. Show that x*+pr> + qx^ + rx + 8 can be resolved into two rational quadratic factors if s be a perfect square, negative, and equal to P^ - H L 19.j4f flr' + 6a;=' + cx + tZ be a complete cube, show that ac^ = dh^\/ and i^ = 3ac. W 20. If ax* + bj^ + cx"^ + dx + e he A complete fourth power, prove bd=\Qiie, be = Gad and cd = 6be 21 /li px? + Zqx^ + 2>rx + s vanish when a; = a or 6 or c, express ^ in terms of ja, q, r^ s^ {\) a + h + c, (2) a2 + 62 + c2, {^) a^ + b^ ^ c' - ^abc. 22. From «« ABC f + (;r-a)(a:-i)(ar-c)(a:-o?) x-a'^ x -b^ x-c^T^Td' prove a" 6» + {a-b){a-c){a-d) {b-a)(b -.c){b -d) c' d? (c-a){c-b){c-d) (d-a){d-b){d-cy 23. Determine the value of « b {a-b){a-c){a-d)'^{b-a){b-c){b-d) + ^ . ± {c-a)(c-b)(c-d) {d-a){d-b){d-cy 24. Determine the value of 0. a» + anal + anal + anal. {a-b){a-c){a-d) t25. li.^-—==a^ + a^x-\.a^'' + a^x^ + ....^ prove a^^ 1, a^^ 2, ^ flj = 3 . . . . and «„ = w + 1, alved into two , negative, and f that ac^ = dfj^ / 1 power, prove b or c, express f ;^ - 3abc. D ; x-d^ i) 233 1. INDETERMINATE COEFFICIENTS. ^26. If J-- = «„ + a^^ + a^ ^ ^^^^^ a^^a^=. a„ , 27. Deterniine a, b, c, d, e so that the n^^ term in tlie expan- ~ (1 _ j.^5 niay be n*c'-». 28. Expand, in ascending powers of x, 1+x ^ sion of 1 1- 2-3.1;' 2 + 3.1;' \-x + ?' t prove ^3 + ^,5, + ^,^^ + ^^^0. ^V + i^.^.... ^ 30. What values of x and y make the fraction, 2z' + (x-u)z + 2b (x - 2c) 3z^ + (fjlb)z + Za{y - 3c) ' independent of z 1 31. Sum the series, 1 . 2 + 2 . 3 + 3. 4 + . . . . n(w+ 1). V' C-32. Sum the series, P + 2' + 3^ + ^^^^ \^ q33. Sum the series, 1 + 3 + 5 + (2n _ 1). )/ )/ ■I) «)■ ' i|- al. S-l, ai = 2, ^ 16 CHAPTER XVII. JJLLM PERMUTATIONS, COMBINATIONS AND DISTRIBUTIONS. 225. The word Permutation is used to designate a number of things arranged in a definite order. Thus abc, acb, bca, hac, cha, cab are the permutations of the letters a, 6, c taken all together. Sometimes we speak of the permutations of a number of things, of which only a part are taken at a time. Thus ab, ba, ac, ca, be, cb are the permutations of the letters a, 6, c, two being taken at a time. 226. The word Combination is used to designate a number of things taken as a whole, without regard to the order of their arrangement. Thus abc, bed, cda, dab are the combinations of four letters a, b,c, d, three being taken at a time. The groups, abc, acb, etc.,' are different permutations, but the same combination. 227. The word Distribution is used to signify a mode of division of a number of things into parcels, or groups. In this connection we shall use the word Parcel to refer only to the things taken together, and the word Group when we wish to distinguish both things and order of arrangement. 228. When we speak of n things without further description ' •^" "-=""^e tnat tiiuy are aii aiuerent, t.e., each is capable of being distinguished by the eye from every other. If ^TIONS nate a number itations of the 3 speak of the nly a part are 3 permutations nate a number order of their >f four letters, I, abc, acb, etc., ion. ify a mode of 3ups. In this 5r only to the in we wish to er description it, i.e.y each is ery other. If PERMUTATIONS. COMBINATIONS AND DISTRIBUTIONS 235 i-o or more aro alike, the exact number of such like thing, «in be speched; and m thi, case each individual thing will be counted .n stating t.o total number. But when wo spLk of « thlgT each of w ,,oh ;n„y be repeated any number of^imes, wo Z^ n d,fferent kmds, with an unlimited number of each kind. Vcrf^neA « n different roays, then the tu,o actio,^ jointly can he fa-formed m mn differed ways. '^ ^t the first action be the selection of one of the capital letters, AJI, C . . . , and the second ;l,e selection of one of the small letters se . Ihe letter A n.ay fl«t bo chosen, and then any one of the ..letters, „ *, o. . . , n.aking n choices in which A is taken first S^mdarly there n.ay be n choices in which B is taken first, aid so on. Thus we have in all n choices repeated « times, il, the two selections together may be made in mn diflerent ways. Con-This principle may easily b„ extended to three or more sets of operations. Thus if the first action can bo perforn.er I m ways, the second in n ways, and the third in p ways, the who e can be performed in mnp ways, and so on to any ex^it 1-230. The preceding Art. contains the fundamental principle of the reasoning employed in this chapter. I„ applying \ to any particubu. problem care must be taken to see thatl^L re J. are d^Jerent and that all the different ernes are included! Ex. 1 In how many ways can two persons be seated in a room containing 10 vacant seats ! .. One person can select any one of the 10 different seats, and :r;oroiTo7ff:.rw:;r"^ -^ '>■» --»'■>« »■ -^'■■«:'» 236 HIGHER ALGEBRA. ; . 1 i di Ex. ^.— In how many ways can the letters «, «, «, «, «7, A, c be arrunged in a line ? Place all the a's in a line with spaces Ijetween them. This can be done in only 1 way. Place the b at one end or in one of the four spaces. This can l)e done in 6 ways. The c can now be placed at either end or in one of the five spaces, making 7 ways. The total number of diderent arrangements is therefore 1 X 6 X 7 = 42. PERMUTATIONS. /r 231. — Tojind the number of jjermutationg of n thingSy r being taken at a tinie. Each of the n things may be followed in succession by each of the remaining n- 1 things, giving n{n-\) permutations of two things each. Each of these n{ti- I) permutations may be fol- lowed by each of the remaining ri-2 things, giving w(n-l)(n-2) permutations of three things each. Proceeding in the same way, and noting that the number of factors in each result is the same as the number of things taken together, we see that the number of permutations, r at a time, is n{n - l)(w - 2) to r factors, n{n - l)(n - 2) (n-r+\), the result required. or Cor.— The number of permutations of u things taken all at a time is n(w-l)(/i-2)....3.2.1. It is usual to denote this product by the symbol In, called "fac- torial n." In modern American works it i;i usually denoted by nl The number of permutations of n things, r at a time, may con- veniently' be wenoweu oy n-if I, «, rt*, A, c bo them. Thig I or in one of he c can now es, making 7 8 is therefore hingsy r being on by each of ations of two ; may be fol- n{n-\){n-1) he same way, It is the same t the number PERMUTATIONS. COMBINATIONS AND mSTRIBUTIONS. 237 232. To find tU number of permutniions of n things, rata time, when each thing may be repeated any number of times. Each of the n things may be followed in succession by one of Its own kind and by one of each of the remaining kinds, giving n^ permutations of two things each. Again. e.ich of these n' per- mutations may be followed by any one of the n things, giving n' permutations of three things each. Proceeding in this way and noting that the exponent of n is always the same as the number of t . -ags taken, we see that n" is the number of permutations required. l-a;. -Seven travellers arrive at a town in which there are 10 hotels; i.i how many ways may they be accommodated with lodg- ings, provided all, or any number of them, may go to one hotel] Each traveller may choose any one of the 10 hotels. Each choice of the first may be taken in connection with each choice of the second, making 10" arrangements for the first two. Each of these may be taken with each of the ten choices of the third, making 1 0" arrangements for the first three, etc. The 7 traveller^ may therefore be entertained at the 10 hotels in 10^ or tm million different ways. I b required, aken all at a called "fac- lenoted by n! me, may con- 233. To find the number of permutations of n things, all at a time, of which p are alike of one kind, q alike of another kind, r alike of another kind, and the rest all different. Let N- denote the required number. Suppose all the possible permutations written out; then, if we place distinguishing marks upon each of the p like things, and permutate them in all possible ways, from each of the original numbers we can form \p permu- tations witliout disturbing any of the other things, givhig in all Nx [^permutations. Similarly, by placing distinguishing marks upon each of the q like things we can form [^ permutations from each of those preceding. In like manner, by distinguishing the I !■!, 238 HIGHER ALOEBllA. r like things, we can form |^ r permutations from each of the lat- ter. But tlie things are now all different, and consequently admit of 1 n permutations. ^x [/> X [^x [r^= [w, Therefore or ir=. L n It Ll IL This process may evidently be continued to any extent. 234. To fnd the number of ways in which n things can he arranged in a circle. ^ Since only relative position is considered we shall get f11 pos- sible arrangements by placing one thing in position and permu- tating the other n - 1 things as in a straight line, giving in all iw-l permutations. These permutations may be arranged in pairs, the order of the things when proceeding from right to left in the one being the same as in proceeding from left to right in the other. If these be considered alike, the preceding result must be divided by 2, giving - [ w-1 as the result required. It may appear to the student, at first thought, that the result should be the same as if the things were arranged in a straight line, viz., [w. If so, place each of the following permutations of four letters, abed, bcdo., cdab, dabc, around a circle, when it will be found that though they are different permutations in a line, they form the same arrangement in a circle. Similarly all the pormutations of n letters may be arranged in groups of n permu- tations each, which give the same arrangement in a circle. The whole number of circular arrangements is therefore \n-^n or \n-\. , PERMUTATIONS, COMBINATIONS AND DISTRIBUTIONS. 239 loh of the lat- consequently extent. 'kings can be 1 get pll pos- 1 and permu- giving in all 1 order of the ne being the er. If these livided by 2, at the result in a straight mutations of when it will ns in a line, ilarly all the of n permu- circle. The EXERCISE XXIV. -1. How many different numbers, each consisting of three fig- ures, can be formed with the nine digits ? -^2. In how many ways can a consonant and a vowel be chosen ^2c %J9 from the word permutationi . ^^' ^i^« l^f es and 5 gentlemen drive out in 5 separate car- i^ riages one lady and one gentleman in each; in how many ways may the party be arranged, including the order of the carriages? l^^6. In hov. many ways may 5 speakers be called upon, (1) pro- viding ^ may not speak before ^; (2) if A must be the third speaker an.I J] may not speak before him 1 ^ 17. By how many different ways may a student go fro^Th^ home to school who lives 4 blocks to the north and 3 to the east from the school-house 1 18. In how many different ways can 6 apples and 5 oranges be distributed among 10 boys, giving each boy one, supposing the apples to be alike, but the oranges to be different ? 19. In how many ways can the letters oi ubiquitous he arranged so that q may always be followed, (1) by ,.; (2) by onlv one^.*; (3)byjusttwow's1 ' ' ' ^ ^ ^ ^ 20. On a shelf are 20 books, of which ft fnrm o .... ,v i— - many ways can they be arranged, (1) keeping the set in order l^' iree at a time, two at a time : the letters of y permutating ;ether;U3) by the letters of taken all to- sibilityl one lady and J may this be I chosen with separate car- r many ways le carriages ? ipon, (1) pro- be the third go from his J to the east d 5 oranges >, supposing litou 8 be (2) by only 6vj in iiOw at in order PERMUTATIONS, COMBINATIONS AND DISTRIBUTIONS. 241 and unbroken; (2) keeping them in order, but allowing other books between; (3) keeping the set together, but not in order? 21. In how many ways can 5 books be arranged on a shelf if any book may be placed either end up and either side to the front ? 22. How many signals can be formed with 20 flags of different colors, not more than 4 flags being used to form a signal, and be- ing placed m a line vertically, horizontally or diagonally ? 23. Tom, Dick and Harry scramble for an apple, an orange and a pear; in how many different ways may they pick them up-? In how many ways if the three things were all alike, or if two were alike 1 ' y C24. In how many ways can 10 persons form a ring so tha^ certain couple may always be beside each other 1 r25. In how many different ways may 8 persons be seated at a round table, the seats being distinguished ? In how many ways may 8 children form a ring ? In how many ways may 8 different beads be made into a bracelet ? .t26. In how many ways can 7 persons sit at a round table so that the host may have the guest of highest rank on his right, and the next in rank on his left 1 c 27. In how many ways can a company of 12 sit at a round table so th8,t the host and hostess may sit opposite each other ? 28. In how many ways can a party of 10 form a ring so that a specified couple shall never be beside each other? 29. On a shelf are placed 5 Latin, 3 Greek, 4 French and 6 English books. In how many ways may they be arranged, (1) keeping those of each kind together; (2) keeping each set in order from left to right or from right to left, but allowing other books to be placed between ? /- 30. In how many ways can the letters of the word syzuyy be arranged, (1) with the three y's together; (2) with "each y separate; (3) just two y's together? iiiiiy ill 1 (Hi ! i ' ' m m u 242 HIGH EH ALGEBRA. I. 31. In how many different ways can the letters in the word indivisibility be arranged, (1) with all the i's together; (2) with just five i'a together? 32. In how many ways can m ladies and m gentlemen form a ring so that no two ladies shall be together ? 33. Twenty male and 6 female candidates apply to a school board who have to fill 10 diflferent situations, 4 of which must be held by men and 3 others by women; in how many different ways may the appointments be made ? COMBINATIONS. ^ ^p235. To find the number of combinations of n things, r being ^ oaken at a time. Let iV denote the required number. Now from each combina- tion 1^^ permutations may be made, making in all iV x \r permu- tations; but this will evidently be the total number of permuta- tions of *i things, r at a time. Therefore OP iVx [r = n{n-l)(n-2)....{n~r + l), {n-l){n-2) ....(n-r+1) N= Ll the number of combinations required. If we multiply both numerator and denominator of tiiis frac- tion by (n~r)(n-r- 1) .... 3 . 2 . 1, i.e., by \n-r, we may write the result in the neat form, In i\^= ^-= — . I r I w - 7' The symbol „C^ is frequently used to denote the number of combinations of 7i things, r at a time. €( -f! in the word i's together; Bmen form a ' to a school fiich must be tny different 'ngs, r being ich combina- X I r permu- of permuta- of tliis frac- e may write number of PERMUTATIONS, COMBINATIONS AND DISTRIBUTIONS. 243 "T.^tSB. The number of combinations of n things, r at a time, is ^ equal to the number of combinations of n things^ n-r at a time. This is at once evident from the fact that when r things have been selected to form a combination, n ~ r things are left to form a corresponding combination. The proof also follows easily from the formula} thus : _n{n-\) ..n-{n-r) + \ n{n-\). .r+\ [^ [ n n-r L n-r \r \n—r\r which is tlie result obtained for „C, '237. To find the total number of combinathns which can be y^piadefrom n things, any number being taken at a time. I» proceeding te farm a combination each thing in succession may be disposed of in two ways, i.e., it may be either taken dr left; and since either m#de of dealing with any one thing may be ffllowed by either mode of dealing with each of the others in succession, the total number of ways is 2x2x2 to 71 factors. But this includes the case in which all are rejected. The total number of combinations is therefore 2" - 1. From this article we get an indirect proof of the following equation : nC*! + «(72 + nCg + .... „Cn = 2" - 1, which may easily be verified for any particular value of n. I 238. To find tlie number of combinations which can be formed from a collection of things of which p are alike of one kind, q alike of a second kind, r alike of a third kind, and so on, any number being taken at a time. nnu^ 4.i,:„, u. j-iitr 'p \jiii.ii'^a iii'txy fjts viispuScu Oi in vt? + i ways, siixcc we uiay take 0, 1, 2, 3 ... .ja of them. Similarly the q things may be dis- 244 HIOHER ALGEBRA. 1 '^ ^ 1 'it J i i id , IN i 1 \ If lr I i ^BaH. 1 ■ 11 ii '^A mSkmS^mul.^ ?!•• „„.™., . posed oUnq + l ways, and so on. The total number of ways of disposing of all the things is therefore (P+mq+lXr+l).... But this includes the case in which all the things are rejected. Therefore the total number of combinations is (p + ^)(q + i)(r+\)....-\. 239. To Jind the number of combinations of n things, r at a time, when each thing nwj be repealed any number of times. Denote the n things by the natural numbers, 1, 2, 3 .... n. Take any combination of r of these numbers (with repetitions), arrange them in numerical order, and to the numbers of this series add the numbers 0, 1, 2, 3 .... r - 1 respectively. The re- sulting numbers must be found in the series, 1, 2, 3....n + r-l. Conversely, from this latter series select any combination of r numbers, arrange them in ascending order of magnitude, and from them subtract 0, 1, 2, 3 .... r - 1 respt actively. The result- ing numbers must form a combination (with repetitions) of r numbers no one of which is greater than n. Hence for every combination of r out of n things with repeti- tions there is a corresponding combination of r out of n + r- 1 things without repetitions, and conversely. The number of the latter combination is (»M-_r--l)(w + r - 2) . n — or j w -f- r - 1 w-1 which is therefore the number required. If we denote the different things by the letters a, b, c, etc., and the number of each found in any one combination by an expo- nent, then if all the combinations be written out we shall have all the terms of r dimensions that can be formed from the n letters. Hence this proposition is often quoted as that of finding how many homogeneous products of r diinP'njtin'no ^««, a^ .c,_4 Jrom n symbols. ber of ways of I are rejected. things, r at a of times. 1| 2, 3 .... w. I repetitions), nbcrs of this ely. The re- . . . . n + r~ 1. bination of r gnitude, and The result- stitions) of r > with repeti- : of n + r - 1 imber of the \ c, etc., and by an expo- e shall have from the n at of findinsf PERMUTATIONS, COMBINATIONS AND DISTRIBUTIONS 245 i 240. The proposition of the preceding Art. may also be proved in the following manner, which will be an instructive exercise for the student : Suppose all the combinations written out. Denote their num- ber by In each combination there are r letters, therefore each letter must be repeated 7* n times in the whole number of combinations. Again, if any one letter, a for example, be removed a single time from every combination in which it is found, the resulting combinations will be those which can be formed from the n let- ters, r - 1 at a time; and the number of times in which a will be repeated in them is r-1 and the a has been removed oiice from each combination; there- fore the total number of times in which a enters into the original combinations is r-1 and this must equal the number formerly found. Equating the two expressions we get n ■■' — ctt r» vi- J VI met. Therefore V r — \ I n n + r-l ^ ML iiii liiii I ffnlii! ,:ii !i f ili! ( t 246 Similarly HIGHER ALGEBRA. W-l = — T—Cr^i, t7-_- = r-1 n + r-3 Lr_ Or-. _ w 4- 1 n ~ ~~2~ ' 1 • Multiplying tiiese equations and cancelling like terms on the opposite side of the resulting equation we get C = ( ^ + ^-l)(^^ + ^-2)....(w + l)»i r(r-l)....2.1 [?i + r - 1 n - 1 /»- 1^ 241. To find the number of combinations of n things of which p are alike, r bei7ig taken at a time. I. Suppose r not greater than j)- From the ;? like things one combination of r like things may be formed; with r-1 like things take each of the remaining n-p unlike things in succession, giving n-p combinations; with r - 2 like things take each combination of two of the w -p unlike things, giving {n-p){n-p-l) LI combinations. Proceeding in the same way we find the whole number of combinations to lie 1 +(.,-.,) , i^-P){n-P-'^) in -p)(n~p-l)(n-p -2) 12 I 3 + etc.y the last term in the series being that in which all the unlike things are employed. PERMUTATIONS, COMBINATIONS AND DISTRIBUTIONS. 247 :e terms on the things of which ike things may the remaining combinations; vo of the n-p find the whole ^ + etc., all the unlike II. Suppose r greater than p. Take the p like things and r-poi the unlike things. This can be done in t^ [t-pY n — r ways, and continue the series as before. Ex. — Find the number of combinations which may be formed from tlie letters aaaabcdef, (1) taking three at a time, (2) taking five at a time. 1. Form one combination from the a' a alone, viz., aaa. Next take two «'s and one other letter, giving 5 combinations. Next take one a and two other letters, giving ~ or 10 combinations. 1.2 5.4.3 I Lastly, without an a we can form ^-^ or 10 combinations. The total number is therefore 1 + 5 + 10 + 10 = 26 combinations. 2. Forming the combinations of h, c, d, e,/in succession, 1, 2, 3, 4 and 5 at a time, and with each combination placing the proper number of a's, we get all possible combinations, 5 letters at a time. The numbers are 5 + 10 + 10 + 5 + 1 = 31, the number of combinations required. 242. Problems occasionally present themselves in which it is required to find the number of combinations (or permutations) of n things, r at a time, when there are several sets of like things to be chosen from. No general formulae can be given for such cases, but the following example will indicate the method to be pursued in any particular problem : Fx. — Find the total number of combinations which can be fornicd from the letters in the word p roporlio n taken 6 at a time. i] n* h iH I I'Hy 248 HIGHER ALGEBRA. This problem must be divided into six parts, as follows: (1) With all the letters different we obtain 1 comb'n. (2) With one double letter we get 3 x 5 = 15 comb'ns. (3) With two double letters we get 3 x 6= 18 comb'ns. (4) With three double letters we get 1 comb'n. (5) With three o's and the rest different we get . . . 10 comb'ns. (6) With three o's and one double letter we get 2 x 4 = 8 comb'ns The sum of these numbers is 53, the number of combinations required. The number of permutations, 6 at a time, may easily be found from the preceding; for the first combination will produce [6 permutations; each combination in (2) will furnish -^ permuta- tions, since there are two letters alike in each, and so on. The total number is 11,130. Jm^ 243. The theorem given in the next Art. usually presents a ' considerable difficulty to beginners; we therefore give a numerical illustration to prepare the way for a general proof. Consider the number of combinations of 10 things taken 1, 2, 3, 4, etc., at a time. We have ^10 10.9 10.9.8 ^ 10.9.8.7 _ 10.9.8.7. 6 10^.8.7.6.5 ' 1.2.3.4.5' ^"-17273.4.5.6' ^^°- Now, observe that each combination is formed from the pre- vious one by placing one more factor in both numerator and de- nominator; and therefore each result is greater than the preced- ing so long as the factor in the numerator is "reater than tho corresponding one in the denominator. In this example C5 is the PERMUTATIONS, COMBINATIONS AND DISTRIBUTIONS. 249 follows : 1 comb'n. > = 15 comb'na. >= 18 coinb'ns. 1 comb'n. . 10 comb'ns. 4 = 8 comb'ns ; combinations asily be found 11 produce i 6 1 -f^ permuta, d so on. The lly presents a ^e a numerical gs taken 1, 2, 9.8.7 JT374'' etc. :rom the pre- rator and de- n the preced- iter than tho uple Cg is the greatest, being greater than any one of the preceding; and of those which follow, g^6» C^= ;=C,, etc., which shows that from C5 each number is less than the preceding. In like manner write out the number of combinations in suc- cession of 9 things, when it will be observed that the numbers increase up to C^j that C^ and C^ are equal; and that these are greater than any others. From these two examples it may be perceived that when n is even, the greatest number of combinations can be formed by , , . w . ^ taking - at a time; but when n is odd, two results, viz., those formed by taking '— and ^ at a time, are equal, and that these are greater than any other. -ffiii , 144. Tofimi the value of r for which tJie number of combina- tions of n thinys, r at a time, is greatest. With the usual notation we have nPr = tfir-l ^ n-r + 1 r Therefore according as i.e., as or as or as nCr >, •-, or <, „Cr.u n-r + 1 r >, =, or <, 1; n-r+ 1 >, =, or <, r; w+1 >, =, or <, 2r; w+1 r <, =, or >, -— . Now, from the nature of the problem, r must be a positive integer; therefore n (1) li uhe even, „C,. is greatest when r= - 17 250 I I HIGHER ALGEBRA. (2) It n be odd, „C^ = „C^_i wnen r-. n+1 , and ill theso two cases the number of combinations is greater tlian in any other, i.e., „C. if greatest when /• = either n- I n+l -__ or -r— , 245. To express the number of combinations of n things^ r at a time, in terms of the number of combinations for smaller values of n and r. The total number of combinations is evidently made up of those which can be formed without including a particular thing, together with those each of which does include it. T^'e number of the former is „_iC^; the number of the latter is „_iC^_i, for any specified thing will appear as many times as different cor*- binations of r - 1 things can be formed from the remaining m - 1 things to place with it. Each of these combinations can be sepa- rated into two parts in the same way, and so on to any extent. The process may be expressed in symbols as follows : »^r — ;i-l^r+ n-l^r-l EXERCISE XXV. CI. How many different parties of Wian be chosen from If persons H In how many of these would a particular person be found 1 '^2. How many different parties of 6 can be chosen from 20 persons ? "^ In how many of these would two particular persons be found 1 "^^ In how many would the first be present and the second absent 1 .3. In Arithmetical Progression there are five quantities con- cerned, and when any three are given the other two can be found. How many different formulae can be given on the subject"? I I PERMUTATIONS, COMBINATIONS AND DISTRIBUTIONS. 251 t 4. From 20 ladies and 15 gentlemen how many different par- ties of .) ladies and 5 gentlemen can he formed ? If the parties \^ considered different when different ladies and gentlemen are partners, how many parties can bo formed ? 5. From a company of 9 men and their wives a party of 4 men and 4 women are to be chosen. In how n.any ways can this be done so that a man and his wife shall not be in the same party? 6 In a basket are 5 apples at two for a cent, and 3 pears a^ cent each. A boy having 2 cents in his pocket wants some fruit How many choices can he make? 7. From 10 different books in how many ways may a choice of one or more books be made ? If all possible choices be made in succession, how many times will any one book be chosen? '" ' ars, 1 dollar, j^' taking one or more of the following sums: 10 dollars, . aoiiar, 50 cents, 25 cents. 10 cents, 5 cents, 1 cent? What is the total value o all the sums thus formed? How n.any sums could be formed by using a 20-cent piece in addition ? C.9. In a basket are 10 oranges, 8 pears and 7 apples. How many different choices of a quantity of fruit can be made the specimens of each kind of fruit being alike ? 10. Apply Art. 238 to find the number of divisors of th.. -.m- bers 540, 720. 11. Find the number of combinations of the letters of the word V ndivisih'ility,i letters being taken at a time. Find also the number of permutations 4 at a time, and the number of permutations all at a time, in which two i^s do not con.c together. S^\ }^' ^'f f""" ''"°'^^' "^^ combinations, (1) 5 at a time, (2) G at ^W, of the letters of the words ever esteemed frieZl J^md also the number of permutations in^ach case. 13. In a basket are 25 oranges worth 3 cents each. How much money should a boy spend so as to have the greatest num- ber of choices ? o «t l^ i iy' I '^r-;-% 252 HIGHER ALGEBRA. \ *■ U. From 15 ladies and 8 gentlemen a committee of 7 is to be chosen. How many of each should be taken to permit of the greatest number of choices ] , 15. There are 17 consonants and 5 vowels. How many from each must be taken to form the greatest number of combinations containing a tixed number of both vowels and consonants? and how many such combinations can be forr.ied 1 IS. In the preceding example how many combinations can b- formed, each containing a fixed number of letters'! What is tb total number of combinations which can be formed, and how many of these will contain at least one vowel and two consonants? ,.. 17. Of 2 H things 7i are alike and the rest are diflforent. How many different combinations of n things each can be formed? 18. The number of combinations of n things, 5 together, ia 3| times the number, 3 together. Find n. -j^- 1». The number of combinations of 2n things, 3 together, is ' If times the number of permutations of n things, 3 together. Find n. 2* The number of combinations of n things, r at a time, is -" the same as the number 2r at a time, and 2^ times the number of combinations, r - 1 at a time. Find n and r. 21. In how many ways may m boys and n girls form a ring so that no two girls shall be beside each other 1 {m >n.) 22. Three aldermen are to be elected from 5 candidates. In how many ways can 4 electors cast their votes, each elector hav- ing the privilege of voting for 1, 2 or 3 candidates? 23. The number of ways of sel^ting x things out of 2x + 2 is to the number of ways of selecting x things out of 2x - 2 as 99 t^7. I Find X. \<'ri 24 From n things, p of which are alike of one kind arid q another kind, how many choices of on© or more things^ y y\ may be made ? 1 PERMUTATIONS. COMBINATIONS AND DISTRIBUTIONS. 253 f, 25. How many different throws may bo made with 2 dice? with 3 dice? with n dice? 2«. In how many ways can a boy select a dozen marbles in a shop where there are 5 kinds for sale ? 27. If (a + h + c + dy be expanded, how many terms will there be in the result? ^ 28. Out of 21 consonants and 5 vowels how many words, each containing 5 consonants and 3 vowels, can be formed ? c 29. Find the total number of permutationa that can be formed from m thin-s of one kind and n things of another kind, taking r of the former and s of the latter to form each permutation. 3#. Find the number of signals which can be made with 4 lights of different colors when displayed any number at a time, arranged perpendicularly, horizontally or diagonally. 31. How many apples must be put in a basket with 9 oranges and 14 pears so that a person wishing to purchase some fruit may have 2,999 choices? 1 32. If n straight lines of indefinite length be drawn upon a plane, no two being parallel and no three passing through the same point, (1; how many intersections will there be? (2) how many triangles ? f 33. If n points in a plane be joined in all possible ways by in- definite straight lines, no two of which are coincident or parallel and no three passing through any point except the original n points, (1) how many lines will there be? (2) how many triangles having their angular points on the original points 1^) how many triangles in all ? (4) how many intersections, exclusive of the in- tersections at the 71 points ? 34. There are n points in a pla.ie, p of which are in a straight line. (1) How many .straight lines can be formed % joining the points ? (2) how many triangles will have their angular points on the oriffinal noinfo 1 35. If there be n straight lines in a plane, no three of which {/ 254 HIGHER ALGEBRA. meet at a point, find the number of groups of n of their point, of intersection in each of which no three points he m one of the straight lines, ^ 36 In a plane are m straight lines which all pass through a ^ven point, n others which pass through another point, and ;, others which pass through a third point. Supposing no other three to intersect in one point, and that no two are parallel, find the number of triangles formed by the intersection of the straight lines. 37 In an ordinary checker-board how many squares can be formed by grouping together any number of the original squares^ How many could be made if there were n squares on each side of the original board ] 38 If a cubic foot were divided into cubic inches, how many cubes could be formed by grouping together any number of the cubic inches without disarranging any of the small cubes ^ How many could be formed if the edge of the original block were n inches 1 DISTRIBUTIONS. 4 246 In the problems which we have discussed in the previous ' sections of this chapter we found two chief elements for considera- tion, viz , the order of arrangement of things in a group and he particular things to be taken to form a group, in both cases the number of things in a group being given. But there are a num- ber of other elements which, when taken into consideration, very much change the character of a problem, some of which are treated in the remainder of the chapter. /..,•„„ The general problem is the separation of a number of things into a series of classes. A very great variety exists in the par^ ticular problems which may be proposed, some of -^^f^l^^^ considerable difficulty. The principal elements on which the dis- tribution depends are live in number, as toliows: 1 The things to be distributed may be alike or different. PERMtJTAtlONS, COMBINATION'S AND DISTRIBUTIONS. 255 2. The classes when formed may bo distinguished from each other by some consideration independent of the elements they contain, or they may not be so distinguished. 3. The order of the things in a class may or may not be con- sidered, i.e., they may form a group or a parcel. 4. Blank groups or parcels may or may not be allowed. 6. Some of the things may or may not remain undistributed. I/. 247. 7'# find the number tf imys in which n things can he divided into tw» parcels containing r and n-r things respectively. A ^^ This is essentially the same as finding the number of combina- tions of n things, r at a time; for whenever a combination of r things is formed, another combination of n-r things is left, and the original number is divided into two parcels. The result is therefore I r \n-r' If r = w-r, and if there is no way of distinguishing the par- cels except by the elements they contain, this result must be divided by 2. ^rB.— Four different books may be equally divided between 2 boys in G different ways; but they can be wrapped in parcels of 2 each in only 3 different ways. 241. To find the number of ways in which n things may be Hvided into three parcels containing r, s and n-r-s things re- ipectively. From the n things r things may be selected in f- I r \n-r 256 HIOHKR AtoEBRA. different ways; and when any one selection has been made the remaining n-r things may be divided in \n-r n-r - 8 111 ways. The two operations may therefore be performed in I w \ n-r [n. \r\n-r \s\n-r-s [^ r j 8 \ n-r-8 different ways. This process may evidently be continued to any extent. lir = 8^n-r-s, and if there is no method of distinguishing the parcels except by the elements they contain, this result must be d" vided by I 3 ; and if two of the three parcels contain an equal number of~thiugs, the result must be divided by 2, as in the former Art. ^a;.— Six persons may be placed in 3 different carriages, 2 in each carriage, in 90 different ways; but 6 different letters can be divided into 3 sets of 2 each in only 15 different ways. '/V'" 249. To find the number of way8 in which n different things may he divided into r distinguislied parcels (blanks allowed). The first thing may be placed in any one of the r different parcels; and when this has been done the second may also be placed in any one of the r parcels, giving r" ways of disposing of two things. Each of these ways may be followed by r ways of placing the third thing, making r* ways of disposing of three things. Proceeding in this way we see that n things can be divided into distinguished parcels in r" ways. Ex. This proposition gives the number of ways in which n different prizes may be awarded to r students. PERMUTATIONS, COMBINATIONS AN1> DISTRIBUTIONS. ^57 fj :50. To find tJie number of vniys in which n dif event things -an he arranged m r different groups (blanks allowed). The first thing may be placed in any one of r different groups- the second may be placed on either side of the first one, or in any one of the remaining r- 1 groups, nmking r+\ different pos.t.ons. Similarly the third may be placed in any one of .4-2 different positions, and so on. Therefore all together there are r(r+l)(r + 2)....(;. + w-l) different arrangements. Ex.-li a lady has 3 different rings, each of which may be worn on any finger of either hand, she can place them on her fingers m 8 . 9 . 10 = 720 different ways K '^}y]'rf^'"^ ^^' ''"'''*''* •^"''^•'^^ *'^ ""'^'"'^ ^ ^^^' things may be divided into r distinguished parcels (blanks allowed). Denote the parcels by A, B, C, etc., and the number of things ^l":' " 't 'I ^W' ^'"' *^'"^ ^^^ ^»^^" g«^ «-h results fs 7 ,: • • •/°' ^^^ ^^^"''^"^ ^'^y^ o^ division. Now, the only restriction is that «4-i + e+ . . . . =,, ,i,ee all the things must be distributed. This is evidently the same as forming all the homogeneous products of n din.ensions from the r symbols. A, B C, etc. The result is therefore r + n- 1 [n \r ~\ Art. 239 Ex.~li n marbles be thrown on the ground to be scrambled tor by r boys, this proposition gives the number of ways in which tliey may be picked up. 252. To find the number of ivays in which n different things can be arranged in r different groups (no blanks). Arrange the n things in a line; we have then to insert r- 1 HIGHER ALGEBRA. points of division among the n-l spaces between the n things. This can be done in L!l:ii. [ 1 In -r ways. The things can be arranged in a line in frt different ways; therefore the required number is n -1 M . r— 1 1 w-r' Ex. — The number of ways in which n cars can be ft Cached to r different engines, one car at least being attached to each engine, is 1 L!^L n Ir— 1 \n — r 253. To find the number of ways in tohich n like things can he divided into r distinguished parcels (no blanks). Place the n things in a row. Since all are alike this can be . done in only one way. Insert r - 1 points of division among the n—1 intervals, and place the things between the successive points in the parcels in order. Tiie result is evidently the number of combinations of w - 1 things, r - 1 at a time; that is. L n-l L'^ r-l n — r Ex. — This proposition gives the number of ways in which n marbles, all alike, may be distributed among r boys so that each boy gets at least one. * The preceding propositions do not, by any means, exhaust the number of problems which may be proposed in this part of the subject; but they contain a sufficient variety for the ordinary cjf ji (1 Qrjt TliOP.ft v/ho dpsire tn pursue the subject further, and to inv*^sfigate the more intricate problems, are referred to the very b 2(> PERMUTATIONS, COMBINATIONS AND DISTRIBUTIONS. 259 Whttrtr'''.M'f 1 " ^'"^^ ^"^ ^'^^"^^'" ^^ William Allan Whitworth which has been consulted to some extent in the preoa ration of this chapter. ^ P*' EXERCISE XXVI. ■ItU" '2° W.77T """ ' '"'^"'" ""o''' •« «J™"y "-ided ^«Pa„lt7 Th^ ™''' "" '' P'"™^ ""^ '""'''^'' -to 3 equal ZZT^ in r '"""'' T" """ *'"^ •'^P''""' ■» 3 different carnages, 4 in each carriage ? - 3. In how many ways may the 26 letters of the alphabet be divided into 5 parts, 4 of the parts containing 6 letters'eachl 4 In how many ways can a selection of (n- \)r things be made from n- 1 sets, containing 2.. 3.. . . .. tilings respec^fvel^k taking r things from each set % P^^^iveiy^fe;. ^ 5. How many signals can be made with 7 differently colored k ^^'^ fhTmTst: r"' ^" *^^ ^^^^^ ^^^^^^ -^^' ^- - --rily^n lifti''. ^'''- ""^""^ ""^^^ '^" ^^ ^^^^^^^* ^^^ be attached to 4 V^ 5 o different engines, any number being attached to an engine ? ^ - 7. In how many ways may 10 apples, all alike, be divided be- "^ " b'l tween 5 boys, any number being given to a boy ^ 8 At a matriculation examination 100 students compete for . If the award is made for general proficiency ? (2) if each scholl/ ship is awarded for one department only ? ^ '" ^ 9. How many signals can be made with 7 different fla^s on ^ c r. niasts, If all the flags must be used and every mast muiTve a ^ " ' ' ^n W ^'' ^T "'T^ ^^^^ '^" '^" ^"""'•^ ^^ *h« ^Iph^bet be made ^ 1 ^"^ ^nto 6 words, each letter being used once, and only once ? ^ I ■ J V A ■ 'y ^60 ' HIGHER ALOEBBA. r 11. In how many ways can an examiner assign 100 marks to^ 10 questions, some value being given to each question "J _ 12. Twenty shots are to be fired. In how many ways may the ^ work be distributed among 4 guns, {\) without leaving any gun ijl unemployed; (2) without any restriction*! \ ^ 13. If n marbles, all alike, are thrown on the ground to be 0-7 "Scrambled for by r boys, in how many ways may they be picked ^ up, (1) if all the marbles are found; (2) if some of them are los^J^ 14. In a ladies' school are 15 pupils, who walk out in 5 rows with 3 in each row. In how many different ways caia they be arranged so that no two shall be twice in the same row 1 15. There are 3n + 1 things, of which n are alike and the rest all different. In how many ways may n things be selected from theml 16. In how many ways can 3 numbers in arithmetical pro- gression be selected from an arithmetical series of n terms? 17. How many different selections of 2n things may be made from a collection of n like things of one kind, n like things of a second kind, and 2n like things of a third kind 1 18. A, B and C have respectively the letters oi proportion, square root and logarithms. In how many ways may they exchange so that each will still have 10 letters, but only one of the persons will have two or more letters alike? 19. A square is divided into 16 equal squares by vertical and horizontal lines. In how many different ways may 4 of these be painted white, 4 black, 4 red and 4 blue, wit>iout repeating a color in the same vertical or the same horizontal line ? 20. Show that m m(m + 1 ) irt{m + 1 ){m + 2) to 71 + 1 terms ^ ^ lil±}l + !i(!i±i)(!^_±l) + .... to m + 1 terms 1-I-T + 1.2 + 1.2.3 ^^^^ PERMUTATIONS, COMBINATIONS AND DISTRIBUTIONS. 261 posed. If 30.me„,bers vote, each for 1 subject, in how many ways can the votes fall 1 ^ 22. Up^ + r things are to be divided as equally as possible among ;. persons, in how many ways can it be done 1 (r = or > M.) 24 In how many ways may the letters of permutations combznaHons, distribution, be divided among 3 per- sons givmg 6 double letters to one person, and 12 letters, no two being alike, to another ? 25. In how many ways may 2n like things of each of 3 different kinds be divided between 2 persons so that each person may have on things ? * / ^ 26. In how many ways may the letters of Ili^^her A Inshra be divided among 3 persons, any number of letters being given to one person ? b b "" 27. The game of },agatelle is played with 8 balls all alike and • TT^t "^J^"^* ^' *° ^^^ ^ '"'^'^y ^« possible of the balls into 9 different holes, each of which is capable of receiving but 1 ball. How many di^erent arrangements of the balls are possible ? 28 If bagatelh/is played with n balls alike and one different and there are y^ \ holes, each capable of receiving one ball, the whole numbejfof ways in which the balls can be disposed .f is ml CHAPTER XVIII. MATHEMATICAL INDUCTION. 254. Suppose we were required to find the sum of the series 1+3 + 5 + .... to n terms. This might be done by the following line of reason- ing: 1. By observation and trial we find that 1+3 = 22, 1+3 + 5 = 32^ 1 + 3 + 5 + 7 = 42. Here it is seen that the sum of two terms = 2% the sum of three terms = 3'^, and the sum oi four terms = 4^. These facts would indicate or suggest that the sum of n terms = n'. 2. To prove that this is so let us assume that 1+3 + 5 + 7 + .. ..(2n-l) = ;i2, and then proceed to examine what will follow from this assump- tion. Adding (2?4+ 1) to both sides we obtain l + 3 + 5 + 7 + ....(2n-l) + (2w+l) = n2 + 2M+l; that is, the sum of (n+ 1) terms = (n-+ \y. 3. It is now proved that if the law holds for the sum of r terms, it holds for the sum of (n + 1) terms. By trial it was found that it does hold ior /our term°, therefore it must hold for Jive terms; and holding iovjive terms it must hold for six terms, and so on. Hence wo conclude that the sum of the iirst n odd natural numbeis always equals n". the series of reason- n of three eta would LS asaump- 1; sum of r tal it was It hold for six terms, .rst n odd MATHEMATICAL INDUCTION. 263 255. Thia ,„otl,od of proof, or line of roasoniny, is called Mathematical Induction, f, its resemblance to tl !t m<^„ of rea.on,ng called " Induction " employed in the discoverlf new truths. I differs from Induction, as applied to the discovery that^tlr ""'T •™""' '" """ ■' ""'™' "» "'- for doubt anJ that the conclusion we reach is no wider than the premises. To a c rtam cla«, of problems, many of them connected with seril Mathematjcd Induction is the only lino of investigation ".d reached by any other method of investigation. Nevertheless here ,s generally a feeling of dissatisfaction in the m nd of h" ^dent when first called upon to reach general results this way weTuZLr Vhtrr '-''' ''-'-'-' '» *"» — •"« - that the aw holds gooS^i, aS'j, aS's are the sums to n terms of n geometric series, whose first terms are each unity, and common ratios 1, 2, 3 Show that ^^ + ^^2+ 2^3 + 3^; + .... (n-l)>Sf„=l"+2'' + 3" + ...,«". U. a-Vhx b +CX c + dx , and if, also, a* = &" = c* = . . . . , a—hx b —ex c — dx then will a^b, c be in G. P., and ar, y, 2 in H. P. 15. A person devised his estate among n persons in the follow- ine manner: A was to receive %P and of the remainder; B, 1 ** 1 S2iP and - of the remainder: C, $3P and - of the remainder, and so on. Find the value of the estate. 16. If the difference between the (n - l)*** and n"' terms of an H. P. be — ;; ^ , find the relation between a, h and c. 1 17. If a;= 1 + - , show that the sum of the series y. 1 + 2a; + 3^2 + . ^.o th terms = ?r, MATHEMATICAL INDUCTION. 267 18. Show that if - + ?^ + ^ - 1 „„j « * c ^ a;2 2^2 2;2 19- ^^-+^=2/+,-=^ + ^, then .2^2,.^!^^^^^^^^^ 20. Show that if ^=2^ + ^+2a,., ^ = .» + .= +2fa and .- = .V/+2<.,, then ^ J/' s' loii ratios jui. win . . n\ — t/ ^^^ • • • • J 22. If P. the follow- and ainder; B, prove that remainder, and 23. If X, ernis of an will each fr md c. l-a^ l_i2 i_^r 21. Show that if ^l±£!i:^'4.^' + «'-^' «^ + i^-c' "^ ^ + ^^ — = 1, then ( 2Ac ' 2ca •i2 + c2-a2\ 2n + l 2a6 + anal. +anal. = 1, a^:t'+ by + 0^x^ = i-a2^1_,.^l_^, ««a^ + iy + c V = a*x' + by + c'z\ qual, and if ^^JLl.^1 ^ (2^ -'^-xf X ~ » then 24. If.(«^^^2^o^+...).(«^,,,^, ..nthen« = . = e = « being the number of the letters. '**■' 25. Show that (x'^ + xit + ',fi\(a'^4.„h^ifl\ A i theform.Y^ + _Ji:F+r2/^^^ +"^ + ^^^ "^" ^« expressed in -;£S£a^ CHAPTEK XIX. ■^ . - - BINOMIAL THEOREM. POSITIVE INTEGRAL EXPONENT. 257. By trial it can easily be shown that (^ + a){x + h){x ■\-c) = 3?-\- X'{a + /> + c) + x{ah + he + ca) + abc, also, (x + a){x ■{■ b){x + c){x -\- d) = a?* + 3?{a + 6 + c + cZ) + x\ah + hc + cd H- da + ac + bd) + x{abc + ice? + cc?a + dab) + a6cc?. From these two cases it is apparent that the coefficient of the highest power of a: is 1 ; the coefficient of the next highest power, the combinations of the second terms of the factors taken one at a time; the coefficient of the next highest power, the combina- tions of the second terms of the factors taken two at a time, and so on, the last term being the combinations of the terms taken all together. 258. The truth of this law of expansion for miy number of factors may be proved by Induction ; but it can reactii^ »»• , shown to be true by calling attention to the mannor in which the various terms of the product are obtained. For instance, when (a; + a)(x + b)(x + c){x + d) are actually mn!'';;plled together, the product is obtained according to the following laws : 1. The whole product consists of a number of partial products, obtained by multiplying together four letters, one being taken from each of the four factors. BINOMIAL THEOREM. 269 2. The partial product x* is obtained by taking x out of each of the four factors. 3. The next term is obtained by taking r out of any three of the factors as many ways as possible, and o7ie of the letters a, b, c, d out of the remaining factors. 4. The next term is obtained by taking x out of any two of the factors as many ways as possible, and two of the letters a, i, ^, d out of the remaining two factors. 5. The next term is obtained by taking x out of any one of the factors as many ways as possible, and three of the letters «, h, c, d out of the remaining three factors. 6. The last term— the one ind pendent uf x—h obtained by taking all the letters a, ft, e, d. Now it is evident that this is only another way of stating that the coefficient of a^ is the combinations of a, h, c, d taken one at a time; the coefficient of x\ the combinations of a, b, c, d taken two at a time, and so on. It is also evident that the same reasoning applies, no matter what the number of factors is, so long as each factor begins with the same letter (x). Consequently we reach the general result that {x + ai){x + a,)(x + a.,),..{x + a„) = a,'« + «;«-!(«, +a2 +,.«„) + x^^-^a^a^ + a^a^ + a^a^ + ... a^_^a^) + . . . a,a^ ...«„, the coefficients bein^- formed according to the law of combina- tion^; that is, the coefficient of x^-^ is the cor.Luiations, taken or<^ at a time, of «i, - .3....«„; the coefficients of x^-\ the combinations cf the same letters taken two at a time, and so on. If, now, we assume «! = a, = ag = . . . . a„ = «, we have {x + «)»» = .r" H- x-\na) + ^n-.|!^^_L)^ j ^ ^„,3 K^-lXn-2) ^3| ^ ^ ^^ =.x- + nx-^a + ^-^IL«- V + !^-lfc:?)^n-3«3 + u U ..a*». 270 HIGHER ALGEBRA. This is the Binomial Theorem ; that is, the law of the ex- pansion of an expression of two terms when the index is a posi- tive integer. In a subsequent chapter it will be shown that the same law of expansion holds for any index. ^- 259. Proof of the Binomial Tlieorem. — The previous result may be reached with much less work as follows : {x + cf)" is the product of n factors, each equal to {x + a). This product consists of terms, each of n dimensions, obtained by multiplying together n letters, one being taken from each factor. For instance, the term x^-^aj^ is obtained by taking x out of (w - 3) factors, and a out of the remaining three. Therefore the coefficient of 3if*-^a\ that is, the number of terms containing ar**-V, will be the number of ways (w-3) thi/ig'i can be taken out of n things, or the number of ways three things caa be taken out of n things ; that is, the coefficient of a*"" V = — ^^ — 1—Jl — Z — L — L^^ Li w-3|3 Similarly it may be shown that the coefficient of a;" L!^ [n~r\r^ Now, by giving r all values possible in this case, that is, 0, 1, 2, 3 .... »i, we obtain the coefficients of all the terms. Therefore {x + af = a;" + C^x''" 'o. + C'aa:"- V+ .... C„a» where Cj, Cn, Cg (7„ represent the number of combinations of n things taken 1, 2, 3 .... n together. Cor. 1. — Write - a for a ; then Cor. S. — Since the coefficients of the expansion are 1> Cl, C/2, Cg. . . . C„, ... y T; _L jrj\»» 1C1 / /. xr3 : / 1\ f the ex- s a posi- that the IS result %). This lined by h factor. c out of ?ifore the iitaining 36 taken be taken is, 0, 1, 'herefore )inations or jf. BINOMIAL THEOREM. 271 Car. 5.— Let .r = 1 and a = ar; then the expansion of (1 + x)" = 1 + C,x + C^c^- + C,r^ + c,x* + .... C,^, Cor. 4.~In the preceding let ar= 1; then (1 + 1)» = 2"=1+C, + C, + C3 + ....C„, 2»-l= C, + C, + C, + ....C,. Thus by the Binomial Theorem we reach the conclusion already obtained in the chapter on Combinations, that the number of combinations of n things taken 1, 2, 3 .... w together = 2"- 1. Cor.5.-In(l+x)"=l + C,x + C^ + ....C„;r"put;r=-l; then (l-ir = 0=l-(7, + C,-.C3 + ....(-l)»(7„, That is, the sum of the combinations taken 1, 3, 5 ... . together = the sum of the combinations taken 2, 4, 6. . . . together, plus unity. ^ 260. Any binomial can be expanded by using the form (1+ar)" For suppose we have to expand (x + y)» this can be expressed in the form, Let - = a ; then (x + yy = x\\+ a)\ We can now expand (l+a)" and multiply each term of the expansion by x^. 261. Since the coefficients of (a + x)» are, after the first term the combinations of n things taken 1, 2, 3 .... ,i together, the coefficient of x is of a^, Yl ra"-'; oix', [Vl zQ ,n-2, Li i\ '- /• Lit ;«**-'. . . . ; and of ar'. I r In r,n-r 272 HIGHER ALGEBRA. Therefore the term involving x^ in (a + a:)" is L" -a"-'■a;^ I r n-r This is called the General Term, or the (r+ 1)*** term of the series. Cor. — The general term of (1 + ar)" is \r \n-r x\ 262. The coefficients of (x + a)" equidistant from the beginning and end of the expansion are equal. For the coefficient of x**~'^a^ is the number of combinations of n things, r together, and therefore equal to L n I r \n-r' it is also the (r + 1)''' coefficient from the beginning of the expan- sion. The coefficient of x^a^~'^ is the number of combmations of n things, {n - r) together, and therefore equal to L: n n -r\r' But it is the (r + 1)'** coefficient from the end of the series; there- fore, since l!^ [1__ \r \n-r \n-r \r' the {r + ly^ coefficient from the beginning = the (r-f- 1)*** coefficient from the end. The important point to notice in this almost self-evident propo- sition is that, since the combinations of n things, r together, = combinations (n - r) together, it follows that the coefficients must be equal when they are respectively the combinations of n things. BINOMIAL THEOREM. 273 r together, and n things, (n - r) together. This occurs when the terms are the (r +!)'»> and the (n-r+iy^ from the beginning, or the (r + !)*•» from the beginning and the (r+ 1)'" from the end— the (r+ !)*»> from the end being the same term as the (n-r+l)^ from the beginning, the whole number of terms being (n+l). Ex. i.— Find the product of (x +l){x + 2)(.r + 3){x + 4). The first term is x*; the coefficient of a;^ = (1 + 2 + 3 + 4) ; the coefficient of a;^ = (1.2 + 1.3 + 1.4 + 2.3 + 2.4 + 3.4); the coefficient of a; =(1.2.3 + 2.3.4 + 3.4.1+4.1.2); and the last term =1.2.3.4. .*. product =a;* + 10r' + 35a:2 + 50a; + 24. Bx. 2. — Write down the coefficient of x"^ in (ar-l)(a- + 2)(ar + 5)(a:-6). The coefficient = {( - 1)(2) + ( - 1)(5) + (-l)(-6)+ (2)( + 5) + (2)( - 6) + (5)( - 6)} = {_2-5 + 6 + 10~12-30}= -33. Ex. 5.— Expand {x + yf. {x + y) ,6_^6 LiL Ta^y + L 2 14 •ar*y Li Li I 3 13 ■^f + Tjj-^^Y LI 1^ Li Li = ar« + 6.r«y + 1 5ar*2/2 + 20a:3y + 1 ftxhf + arein A. P. Find w. / 25. For what values of n are the coefficients of the second, third and fourth terms of the expansion of (1 +^)" in A. P.? •1 I BINOMIAL TUEOREII. 277 «. 26. Simplify {x+Vf-\y + {x- s/y' - 1)« -*, xr bhe id. I 27. Simplify ( ^m'^ 1 + \/m^-\f - ( \/m'-^ 1 - ^/^i^^~\)\ ^28. If a be the sum of the odd terms, and h the sum of the even terms, of (1 +x)", show that (1 -x^Y = a?-l^. A 29. Simplify (5 V2 + If x (5 V2- 7)\ (2 1 \' -x' ) . 3 ixj X ) . C32. Find the coefficient of .i'" in the expansion of (^^ + — ) • 33. Find the coefficient of .i'" in the expansion oi (x^ + — \ . ^ 34. Find the value of - - 3w(n-l) 4w(m-1)(w-2) , ,,, l + 2n + -^—-^ + -^ -^ -^ + ....(w+l)l when w is a positive integer. 263. ^o ^w(/ ^. , =, <, r*"' term J. n-r +1 X according as — . - >, =, <, l- a eras or as or as or as r i a n-r +1 r n+ 1 a a >, =, <, 1 + -; r n + l X >, =, <, r. n + l If is an integer, the (r + 1 )»" term = the r*" term when 1+- n+l 1 + a X and theee will be the greatest terms of the expansion; for any greater value of r will make n-r + 1 X <1. 7^ i ■\ i BINOMIAL THEOREM. 279 n+l . If — — • is not an integer, but has for its integral part m, then 1 + X n-r + 1 X r a cannot be > 1 for any value of r > w; that is, the (»•+ 1)*" temi, cannot be > r*** term for any value of r > m. Therefore the (r + l)*"* term > r'^ until r = m; .*. the greatest term is the (w + 1)* term. N.B. — The student should observe that this proof applies solely to the numerically greatest term, and therefore applies to (a -a:)". '^ j""^^^ ^' — FiJid the greatest term in the expansion of (1+ar)" f when 2 ar = - and w = 6. o The (r+ 1)*" term is >, =, <, r*" term, n — r + 1 according as or as But .'. according as or as • * ^> — » *^i ■'^ J 71 + 1 >, =, <,r. 1 + X w = 6 and x= - o _2 1 + - >, =, <,r; 14 The greatest value r can have in order that — may be >r is 2 ; therefore the third term is the greatest, and its value is 6x5 ./2\' _ 4 6x5 ./2\'' ,, 4 ,., Ijj 280 HIGHER ALGEBRA. ^- Ex. 2. — Find the greatest tsnn in the expansion of (a fa:)" when *=Q» ^~7» '* = 8. In this the condition becomes or w+ 1 1 + a X 1 + 3 27 >, =, <,n >, =, <,r, >, =, <,!'. Therefore the greatest value of r is 3, and the fourth term is the greatest, and its value is, 8x7x6/lvVlV ^^ 1 1 lV2^i3JU>'^^^''2i3^64- 265. The following examples are suggestive, and worthy of attention : , 'Ex. 1. — Find the sum of the series, that is, find the sum of the squares of the coefficients of (1 + a;)^ (1) /-I \« 1 W(W - 1) o {\+xY=\+nx + -^ — -x^ + a;"; ahio, \1 {x + 1)» = r" + na;»-i + ^ , , V "^ + .... 1. Li (2) If, now, (1) and (2) be multiplied together, and the coefficienl,a of x^ be collected, their sum will be the given series. BINOMIAL THEOREM. £81 But (1) X (2) = (1 + x)- X (x + 1)» = (1 + ;,)2.. ,^/rf°!l*^^ ^'''^" ?"^^ """^^^ ^ ^^"*^ ^ **^e coefficient of ;r» [2n in(l+ar)*»; t.e., equal to Hence 1 + 1'= 2n [2^ ^a;. ^.— If Co, Ci, Cj C„ denote the coefficients of (1 +«)» find the sum of ' 2 3 C^o + "o" + -^r + . . . . Now /7j-^^a.^2, cr 1 . ^ n+V n n(n-l) Multiply both sides hy (n+l); ... ^(n + l) = (n+l) + (!L±i)!?^Mn)^^^^ 11 Li Add 1 to both sides; /. >y(n+l) + l = l + (^+l) + 6i±lK^^.....l^(l^lj.+,^2''+^ LI .-. ^=2»+»..l4-(w+l). EXERCISE XXIX. Find the greatest term in the expansion of; 04. (a + 6)20, when a = 2, i=3. *^ C^. (2x-yy% when x^i, ij = 5. ^ C3' (^+1) ' when x=e, y = 8. . . /2x 3a\« , Q«- (-3-+:^ ) » when x=9, a =16, 19 <0 * ^T' /^ / ■I 282 HIGHER ALGEBRA. ^^^ Find the value of the greatest term in the following: jk6. (1+ar)*, when *= 3' ''* = '*• , 6. (2-3a;)», when «= g^* ^ = ^- ^ 7. {a-bf, when 0= 2, 6-3, r = 4. 8. («+-)» when 05=2, n=»8. '> ^ 9. In the expansion of (1 +«)*' the coefficients of the (2r + 1)*^ and (r + 2)*'' terms are equal. Find r. 10. The second, third and fourth terms of {a + rY are 240, 720 and 1080 respectively. Find the values of x and n. ell. Find the relation between r and n in order that the co- efficients of the fifth and (2r + 5)"* terms of (1 + x)" may be equal. ^ 12. If the coefficient of the Sr"* term from the beginning of (l+x)^ equals the coefficient of the (r-l- 2)*'' term from the end, find the relation between r and n. If ao, ai, aj. . . . a^ denote the ioefficients of (1 +«)*, prove: --13. ai + 2a2 + 3a3 + ....wan = »i-2"-- 11 ^n ^ ^ cl«. Oo-2ai + 3a2-....(-l)"(w + l)an = 0. ^ ,;^17. — + — + + .... = ^• ao *i ^a a— >i.-i 18. (Oo + «l)(«l + «2) • • • • («n-l + «») '- a-^. . . . «n(^+ ■"•)" l!L ^19. ai-2a2 + 3a3-....(-ir-'wa^ = 0. 20. a^ + 2a, + 3ffj + . . . . {n + l)a, - (« + 2)2-\ /' /^ ^ 03 II BINOMIAL THEOREM. > 283 ^ C21. a, + 2a3 + 3flr, + . . . . (n _ i)a^ = 1 + (^ _ 2)2»-i. 22. a^, + a,a,^, + .... a^_^^ = U= . ^7^^^ ^ " ^ ^ r I n - r I n + r / 23. aotti + fljOj + . . . . a^_iO„ = — [2n ^rA^/v A/ v; ^^^ |n+l | n-l • ~7/j^ ^ < ^' v; = when n is odd. ( 2 ) luoxi^ I i< ^m'a^ tc^tlc ^q^v»o <-t/6^ j^J-^Cuo . > CEAPTEK XX. i BINOMIAL THEOREM. ANY EXPONENT. n^266. In the preceding chapter we found the form that the expansion of (a + x) assumes when n is a positive integer. This was easily obtained, since (a-f x)» was taken as the product of n equal factors, and therefore its coefficients came under the law of combinations. We have now to prove that the /or»t of the expansion of (a + xf is the same when n is not a positive integer as when n is a posi- tive integer. (^ ca' 267 By actual division we find that 1 (1+^) - or (l+a;)-2 = l-2a; + 3a;'-4ar' + .... ,^(-2),,(-2)( -2-l) , " "IT — [? — Hence we see that in this particular case the formula holds good. Similarly, by actually extracting the square root of 1 +.r it can be shown that (1 +«)' = !+ 2^ + 1 K-.-).i(H(H Li [3 '•1/ T • • • • 268 We now proceed to prove that the law holds good for all values" of n, fractional and negative. At the outset it will be BmOMlAt tSEOREM. 285 tiecessary to call the attention of the student particularly to the fohowing statement, the truth of which he must be convinced before he can understand the proof of the Binomial Theorem for any exponent. ^\ FORM 69. If two algebraic expreasiona are multiplied together, tlie FORM of the product is independent of the value of tJie letters involved. Thus (a + b){a -b) = a'- b\ no matter what values may be given to a and b. So, too, if {% + axX + a^ + .,..a^x%b^+b^x + b^ + .. . . i«ar») = ^0 + ^iiar + ilj^ + . . . . ^j^a*», the fr«TO of the product is independent of the values of a,, i„, «i, Ai. . . . Of course, the valv^ of the product will change, but Its algebraic expression will remain the same. The application of^his totjie proof of the Binobial will be seen in the following o provk the Bitiomidl Theorem when the exponent is a positive fraction. Lrt. 27o: Let the series, , . mlm - 1 ) „ Li a> be denoted hyf^m); then the series, T , n(n - 1) „ will be denoted by/(w). Li (2) By the reasoning of the preceding Art. the for m of the product of (1) and (2) will be the same, whatever values be given to m and n. But the series (1) i? the expansion of (1 +a-)'» when m ■=-!«?! 286 HIOH&R ALOKBRA. is a positive integer, and (2) is the expansion of {\+xY when n is a positive integer. Therefore the product of (1) and (2), when m and n are positive integers, is {m + 7i)(m + n-l). ,„, (l+ar)'"+" or \-h{m + n)x + ^ -—^ ar' + {^6} Hence (3) is the form of the product of /{m) and /(w) for all values of m and n. Also, since y1^m) denotes ( 1 ), J{m + n) denotes (3), and therefore also, f{m)>^f{n)%f{p)=J{m^n)y^f{p) =f(m + n + p). Similarly it can be shown that /{m) xf{n) xJIp) to w factors =/{m + n +p . . . . to » terms). Now, let ^ ' in — n—p = . • • • T» where h and k are positive integers. ••■/(I) x/C^)- •■•*"* *^*°''' =/(^ + I .... to A terms), /('i)=(i.4 or But, in accordance with the notation eK^ployed, 'h\ . k"^ k\k ) L 1 li vl , X''" k\k V »C^ "t" • • • rj I tl (1 BINOMUL THEOESM. 287 Let J ^n: then k 0\it -I n(n — 1 ) , Li. that is, the Pmomial Theorem holds good when n is a positive ([vvA. r^**^ (Kur^ TV) prove tJie^inomial Theorem for a negative index. Since /(w) x/(n) =/(w •+ n) for all values of m and n, let n = - m (w being taken positive). Then ^Iw) x^i; - rn) =y|;»n - m) =^(0). But/(0) = 1, as it is obtained by puttingm=0 in the series, , mlm - 1) , Li ••• y(»»)xy]:-m)=i, or But ^-'">=;t)=(iT^=a+^)-" (1+^) --i+zl».^(-"'X-'»-i)^^ Li Li since j^- m) stands for i +<~'">t I <""*>< ~ *"- ')j' |.. .. Thus it is seen "that the Binomial Theorem holds good for the Tiegative index, - m. >^4^272. The proof contained in the two preceding Arts, presents >r one difficulty, which needs some explanation. It has been stated thaty^m) ><.J\n) =/(m + n). Now, what meaning must be attached , to such a statement when the series which yj^w) and^^n) represent ^^AJafi^^e divenient ? Such series are the expansions of (1 - x)-^ and (1-a:)-^ when ar^l. (See next Art.) \VTien x<\, (l-a;)-« is aesB I ^Ht tilOilEIt ALOfifiRA. arithmetically = 1 + 2a; -^ 3x» + 4x» + ex:, an id^l - x)-^ is arith- metically «=. 1 +«+ a:* +«'+ . oc and .-. (l-ar)-»x(l-af)-'or (1 - ar)-» is arithmetically = 1 + 3x + 6x» + cc, which is the productof (l+2x + 3x» + 4a:» + .... oc)(l +ar + a:' + . . . . oc). But when a:"" 1, (1 -ar)"' and (1 -^)"^ a>"e not respectively arithmetic- ally = 1 + 2ar + 3x» + 4x» + oc and l+x + x^ + a^ + ....oc, and we cannot assert that (1 + 2a; + 3a:»+ . . . oc) x (1 +ar+a;--l-ar»+. . .oc) x» 1 + 3a; + Gar* + oc . We can, however, assert that the first r terms of the product of (1 + 2a;-l-3a;» + . . . oc) and (l+ar + x» + . . . oc) are the same as the first r terms of (1 + 3x + Gr* + . . . . oc), and, generally, that the first r terms of the/(w) x/(n) ftre the same as the first r terms oif(m + n). 273. It has been stt ted in the preceding Art. that the expres- sions, l+2a; + 3a;» + oc and 1 +a; + a;* + a;' + oc, are, when a; < 1, the arithmetical equivalents of (1 -a-)"' and (1 -a;)-^ Thia can be shown by summing the series according to methods already employed in the chapters on Progressions. If, however, a;~l, 1 + 2a; + Sa;* + oc and 1 + a; + a;^ + oc are not the arith- metical equivalents of (1 - a-) "^ and (1 -x)-\ This can be easily proved as follows : For, if possible, let (1 -a;)"* = 1 + 2a; + 3a;'' + 4a;3 + . . . oc . Then if these expressions are identities they must be equal when a; = 2, in which case (i-^r=(i-2)-^=(-i)-^=(Tri7=i. and l + 2a; + 3a;» + ....oc = l+2.2-f-3.22 + 4.2'-|-....oc = oc. That is, 1 =« oc , which is absurd. The real value of (1 - x)-^ when a;~ 1 is ~_ v actually divided out, gives (r + Dx'' l+2a;4-3a;2 + 4a;' + .... Vi ^• (1-a;) -5, which, when ^ V r btt^OMiAL THBOR&M. ^dd If ara + 1 ,\>a < n+l> r a; n+l > 90 { in by be th BINOMIAL THEOREM. 295 Since w is a fraction, {j~- - l] can never be made = 0; and by taking r great enough (that is, by taking a sufficient number of terms), /— Ij can be made as near (- 1) as we please, and therefore ( 1 (^-)^ can be made as near — as we please a *^ In this case whether there will be a greatest term or not depends upon the value of a' If - _ 1 after a sufficient number of terms have been taken, each term will bo equal to or greater than the preceding, and there will be no greatest term; but if -<1 a ' there will be a greatest term, and it will be found from the con- dition that — — >r. As in Art. 264, the greatest term will be 1 + - It — — IS not an integer, the greatest term will be the (m + 1)***, where m equals the integral part of • the (r+ 1)*" term when r = ^^-^~-^, if r is an integer X 1 + a X Then III. Let n be negative. If n is negative let it = -7n (where m is positive). n-r+l\ X /-in-r + l\x / n-r+l \ X ^ /-m-r + 1\ x _ /m + r-l\ x \ r J ' a \ r )a~ ~ [ '^ / ' « ' As what is required is the numerically greatest term of the expansion, the negative sign may be neglected, and the multiplier will be [m + r-\\x ^, , > / j _ . Then the (r + 1)*" term = r"' tern; 296 HIGHER ALGEBRA. according as or as /m + r-l\x_ V r )a^ As in the preceding case, if mis fractional, { — l-lj 'm — 1 may (m-\ .\x be made as near unity aa we please, and .*. ( "^ ) a *^ ^^^^ - as we please. If - "^ 1, there will be no greatest term; but if <1, the greatest term will bo Tound from the condition that X a fm-l ^\x or or m— 1 a - >- -1, r X m — 1 "-1 X <>r. m—1 The rest of the proof is the same as in I. and II. when is positive. If however, — ^ is negative, a new case arises, for then ' r /f^ ~ ^ + 1) is always < 1, and y^^—- + A\ ^^ always < 1 if x~ a. Therefore the successive terms of" the expansion will each be less than the preceding, and therefore the first term will be the greatest. If x>a the greatest term will be found as in I, J^nd II, BINOMIAL THEOREJL 297 Ex. 1. - Find the greatest term in the expansion of (a + xY when n = — and 4a; = 3a. Here the (r + l)'** terra is > the r^^ as long as or -' 4 I r J 21 2 ■ r "-^3' or 21 7 9 Therefore the greatest term is the fifth. -1 ^a;. 2, — Find the greatest term in the expansion of (1 «)"" 3 3 when M = 2 and x=- . 4 The (r + 1)*'> term ^ r*"* as long as ^2 +r-lM / 2+r-l ^l>^ / . 2a3 n or 1 r. Therefore first and second terms are the greatest. EXERCISE XXXI. Find the greatest term in the expansion of: 8 1. (1-ar) =■ when a;=-. 2. (1-a;)-'^* when ic=. -. " 4 90 298 UIGUER ALUEBRA. QS. (l^•a;)-" when x=- (1 ~ a:) • when ^ = ^ • C 5. {1 + ar)^ when x=-. f^. {2z + by^ when ar = 8, y = 3,, C^. (3a;2+52/')-'' whenar=9, 2/=2, n=15. Find the first negative term in the expansion of: ^- iy+l^f' ^10. If cTi, aj, aj, or^ be ary four consecutive temrof an ex- panded binomial, prove tha^- flj + ffj ttg + O4 0^ + «3 SPEriAL APPLICATIONS OF THE BINOMIAL THEOREM. 1. Find the sum of the coefficients of the first (r + 1) terms of (l-a;)« Let ao, a^, aj, aj .... a^ ... . be the coefficients of (1 — x)**. Then (1 - rr)" = a^ + a^a; + ('2^'^ + <^3^ + «r^'' + ; (1) also, (l-a;)-*=l+a; + a;Har3 + x^ + (2) By inspection it is seen that the coefficient of x^ in the product of (1) and (2) = »(, 4- (Ti + O2 + • • • • «r' ^^^ the coefficient of ar»'in the product of (1) and (2) = coefficient of a;'" in (1 - a')"(l - a:)"^ or (1 - a;)""*, and the coefficient of a;*" in (1 _ ^)n-i ^ (n-l)(n-2)....(n-_r)^ _ ^^^ (n-\\ln-1\ (n-r) ,, ,*. ao + ai + «2 + «r = -^ — j-^ ^ -(-!)''• BINOMIAL THEOREM. (1) (2) 299 Ex. — Find the sum of the first (r + 1) coefficients of (1 - x)-\ Let (l-^)"' = ao + «iar + a2x' + ....aX+...., also, (l-ar)-»=l+ar + ar» + .... ;. flo + a, + oj + of^ = coefficient of .c'" in (1 - x)-\l - z)-^ or _>.V-4_ '*-5-6-...(r + 3j |(^ 1.2.3.4.5.6....(r + 3) (l-xV Lr. ^ 1-2.3 ... r _ (r+l)(r + 2){r + S) 2. Find to five places of decimals the value of V98. Vr8=Vioori=J,"oo(i4„) = iojrj=,o(i-l)^ .ikl!l /i\' 2(2-0(2-^) /iv 1 LI 'W [3 [50) -^""j = 10(1--? ! I__ 1 1. 100 8x2500 16x(60)'~*"7 = 10(l-J— i L__ \ t 100 20000 2000000"'" 7 = 10 - 1 1 10 2000 200000' To obtain the values of the several fractions as d ^oimals we proceed as follows : 1 = -1 10 ^' 1 1 J 1 2000 2 ^1000" 2 = ;; X~:rTr= -(-001) = -0005, w 17 I '3 300 HIGHER ALGEBRA. i /. 10-T7^- 200000 100V2000, 1 •000006. 10 2000 200000 )ooj = 10 - (-100505) = 9-899495. Similar artifices can be applied to find the values correct to a given number of decimal places of such expressions as V^126, i?'2400, e*;c. ^ 3. Show that the coefficient of 3^^ in the expansion of is 2n. (1-^)' = (1 + 3a:2 ^ 3^ ^ ^8)1 1 ^. 2aJ + 3;i;« + 4a-« + ....(/+ IXr*)-- + ... .}. The coefficient required consists of all the terms in the product of these two expressions containing n^'^ ; and since every terra of the expansion of (1 - 3?)~^ contains powers of a?^ and only one term of (1 + x'Y, viz., .^^ contains a power of a-', therefore the coefficient required = (coefficient of a:^" + coefficient of a:^""') in {\-3?)-\ The coefficient of r*" in (1 -ar')-2 = w+ 1, and coeffi- cient of a-^"~* or a;*""^*' = w - 1. Therefore coefficient of a?"' in pro- duct of {\+xyi\ -ar»)-2 = n-H 1 +w- 1 = 2w. #- ..^1 4. The following are illustrations of approximations : Ex. 1. — If X be so small that its square and higher powers may be neglected, find the value of >^0) (l-7ar)*(l + 2a')-*. j^xpanding and neglecting powers of x higher than a;' 23 f49 7 49 3 21 (l_7ar)«(l + 2ar)-* = (l-^ar--a;'-j-....)(l--a: + _a:^-....) =.\-^x^—^^ 23 = 1 - -77-^, ncsi iy. Ha\s BINOMIAL THEOREM. Ex. 2. — If p q he small compared with p or q, prove Jp ^ (n-t-l)/> + ( n 3jiJ9'_ \/(!^)2«-3 + ... . v^!^J^il^^.V^^^-^^^i«^feii^te%*»«ri»*^^ 304 HJ0H6R ALGEBRA. Ex. 1. — Find the sum to infinity of 2«/l-n l::5_L,^(^+^)/^-^\'' n{n+\){n+2) (\-xy By inspection it is seen that = A + — V"- /"-!_ V"- (^ + ^>'' 1 +0? ^ ^ ' ^a;. ^.— Sum to infinity, 1 + ? +-M +-?J^^ + .. .. •^' 6 6.12^6.12.18^ Arrange the series as follows: 2 5 2 5 8 -i^-G-)^v-a) 3 3 /W 3 3 3 /1\' By inspection this is seen to be the expansion of 9. The sum of a series is frequently found by observing that it is the coefficient of some power of x (or other quantity) in a series formed by multiplying together, or otherwise combining, two or more series. Ex, — Show that if w be a positive integer not less than 4, , , 4.5 wCn-n 4.5.6 w(w-lVw-5^ u 2 Li .1 Binomial theorem. (l+x)-*=l-4^ + ii.^2_4.5.6 and [1 Li •*- "f" • . . . 305 (1) It is evident that 1-4^ + 1:^. ^zl)_ [2 1^2 = the coefficient of x' in product of (1) and (2) = the coefficient of ar<* in (1 +.)-.(! + 1)" or (1 +«)»-* X" But every terra of ^ / .'. the series = 0. a-" contains x (with a negative index); "" ind the number of homogeneous products of r dimen- can be obtained from n letters. Let a, b,c, d.... be the n letters. Take n series, l+aar + tfV + aV + ....oc, l + bx + b''x^+b\r' + ....ac, 1 + CX+ c^c^ + c^.r' + . . . . oc. in which ax, bx, ex are each < 1. It is evident that if these n series be multiplied together, the coefficient of x will be the products of one dimension obtained from a,b,c,d...., the coefficient of x"" will be the products of two dimensions, and, generally, the coefficients of x^ will be the pro- ducts of r dimensions from the given letters and their powers. 306 BIGHER ALGEBRA. I The problem, then, is to find the number of terms forming the coefficient of .t;'" in the product of Now, l+ax + a^x^ + a^:^ + ...,^ = ^i_a^y^ when ..rr .^i_J7. Show that the coefficient of x^"' in '^ is 2m + 1 k^ {i+x+x'^y ' X 18. Find the coefficient of .c'" in the expansion of ,^<* (1.2 + 2.3j; + 3.4.c2 + ....oc)2. 19. Show that the coefficient of x"" in the expansion of 2^^*"^ (1 + ;r + 2;r2 + 3:r3 + . . . .)Ms ^l^-iy . 6 20. Show that the coefficient of x" in the expansion of - 1 -. V^ i+x + x^ 9 is 1, or - 1, according as w is of the form 3m, 3m- 1 or Stn+l. **'*«*»*»SVSW«*l»fa«j. ! ! 81Q HIGHER ALGEBRA. CL21. Prove VI =1+^ +h2,MlL, V^ ^2. Prove^/^=l_l.l .U 1 _1:3.5 1 V V. >/3 2 2 2.4'22 2.4.6'23"*'"" ^ um the series, ^Vi^l! + i:llLll^ 3. 6^3. 6. 9^3. 6.9. 12'*'-"- 1 1.3 1.3.5 'rove - H + _ + rv- ~ 1 / 4^4.6 4.6.8 ^••^^• .A H) .V.?/1A' 1 '3 /3.5 3.5.7 'OVel+.;r+ +__!_L1_. ,. o 8^8.10^8.10.12^'-"°*= 2- Pr„vel+i2+lLy^l;4Li|^....„,12, L>^ \ '>;' ijW ,^* '/ 14 14. 16"^ 11.16. 18 AJ7. Prove 1 + ?^ + 2n, (2^ + 2) 2n(2n + 2)(2n + 4) ^ ^^ 3 3.6 ^ 079 + •••• ^ I 3"^ 3.6 ■*■ 37679 +""J^ 0^8. Prove T^/l + - + !!i!!zl) . Hn-l)(n-2) >^ ^ ^ 7 7.14 ^7.14.21 ■+••••} ^ = 4«/i + ^ + !!(^±l) , !»(!M:i)(w+2) ^ ' 2"^ 2.4 ■*" 274~fi +••••}• {^9. Prove ^-^_Ji_ l+n.^+!!(^-tl)r_^ a" b-a ' |2^ 1 {b~^y+-" -=(-i)"r». y C^O. Show that if a; > - i, y' —5_ = -^-.-^/-MM:3/^\3 1.3.5/ ^ ^4 Va;+1 !+«' 2Vl+ary ■^2.4^1+^/ '^27r6[urx) '^"" i II < \ \^<' ... Y + . . . .^, +....}. Y l" • • • • BINOMIAL THEOREM. 31 J ^1. Show that (1 + :r)2» = (1 + ;,)« + ^1 + ^y., ^ ^ ^±i^:r^(l + ;.)n-2 "^ L_ g42. Show that if a < 6, 33. Show that la-]-bY{--i.t^^ «* 4.5.6 a' ^ '\b'' 1 />«'^i.2'i*"r72T3'P"*"--'7' / l+ar [2(1+^)- " + = 0. one- A^i. Show that if the numerical value of y be less than third of that of a-, [+n|'-^U!^±lVJy-\^ ^(^ + l)(^ + 2)/ 2y y V+y) [2 U+y; + jT (^j+--.. 35. Prove {n? - 2)* =^!zi|i_i.__J 1 1 1 1 n ^ 2 {n'-lf 8'(rt'^-l)* 16 ' (^^2Tri)6--- ••}• 36. Prove that if n be an even integer, \ 1 1 1 2"-» "" \7^ ^•^ + -... = 2'-(l+r). r 312 HIGHER ALGEBRA. „^ _ 1.3.5....(2r-l) Pin+l +PiP2n +PiP2n-l + PnPn+l = S ' 39. lipr = 2ni ^, prove ^{P2n -PlP2n-l +P2P2n-2 + - •'•(- l)'->„-i/>n+l} =Pn+(-'^Y'W- 40. If ffo, rt,, flj ^„ are the coefficients in the expansion of (1 +;r)" when n, is a positive integer, prove (1) «o-«l + «2-«3 + --.-(-l)'"«r = (-))'" , ,^ , . ^r \n- r -I (2) ao--2ai + 3a2-4rt3 + ....(-l)''(n+l)a„ = 0; n (3) flo^ - ai' + a^' - «3' + . . . . ( - 1)X' = or ( - l)^. 41. Prnv. 1 ■.■'^. + ilj.^(^-i) , 3-4. 5_ n(n-l)(n-2) li 1.2 [2^ '1.2.3' = 2''-3(w2-:-7m + 8). , - 1.3.5....(2r-i) 5.7.9.. ..(2r + 3) + ... 2.4:6.. ..2r """ ^'^-2X6. ...2r ' ^'^"^^ 1 2' 1 + 43. Prove n{n-l)]\ /lY(n(n-l)(n-2)Y i-r-G)FLi^T^a)r-^^^^ir^T--- _/7y. ri(n-JL) /6\ n(n- l)(n - 2)(n- 3) /6x = 44. If a^ be the coefficient of a?*" in (1 +x)'*, prove that if /5; be less than n, [n_ \n-l \n-2 \ n-k "^|n->fc-l'^''''[glT:72 -.... = 0. prove BINOMIAL THEOREM. 45. If.r, = a.(.r+l)(^+2)....(^4-n-l),8howtha' (This is Vandermonde's Theorem.) 46. Prove that if H + r^" - /. j_ ^ ^ ■ « mu ii{^L+x) -c^ + cix + c^^ + ,,,, c^^n^ then n(l + x)-^ = c, + 2c^ + 303..=^ + . . . . ^„^»-i. 47. Prove, using the notation of the preceding example, that and Co2 + 2c,2 + 3c22 + ....(^+i),^2^ 48. If/(r) 813 - + n [n n(n - 1) \n iLiHizz TEr]ZIZZl''ni~'IEin£EZE ; + . then/(0) + n/( 1 ) + !^^) /y 2^ ^ /^.) - (^^ ^ 1 )(2^ + 2 ) _3n^ 49. W(l+a:)« = ao + ai.r + ^2.r=' + ...., then will < + 2a,' + 3a3' '.f^..naJ_^ ~~a7+«7+«7+ . . .T<^?~ ~ 2 ' ^ '^^^"^ ^ positive integer. 50. Prove 2" - (74 - l)2"-2 L ' 51. Prove 2».H^~— ^ on-2 . w('*-1)(w-2)(m-3) \2n V V.2'' •2"-* + .... l!iL w 52. Prove that the sum of the first n coefficients of the expan sion, in ascending power of x, of (l-.r)^ " IS On-4 ;^i n being a positive integer. m 314 HIGHER ALGEBRA. 53. In a shooting competition a niah can score 5, 4, 3, 2, 1 or points for each shot. Find the number of different ways in which he can score 30 in 7 shots. 64. A man goes in for an examination in which there are four papers with a maximum of m marks for each paper. Show that the number of ways of getting half marks on the whole is ^(m+l)(2m2 + 4m+3). 55. There are two regular polyhedrons marked in the manner of dice, and the numbers of their faces are m, m + n respectively. How many different throws can possibly be made by throwing them together 1 56. If, in the preceding question, the number of polyhedrons be four, and the numbers on their faces 3, 6, 8, 12 respectively, show that the number of different throws that can be made by throwing all together is 552. 57. lis = a^ + b\ p = 2ab, P = (a + hy, show that P. P*.pi.P*....< .oc L 2 n{n- 1), »^-=' + , 68. !£/(», ™) = i-„(-±-) +!!(!!--JL)(_i ) _ ,h„, m \m+pj 12 \m + 2jo/ ' that /{n, m) = \~\f{n - 1, m +p). 59. li z^ + z + l-=0, show that the sum of those terms of the expansion of {l+x^, in which the index of a: is a multiple of 3, 60. If a,. = coefficient of a;'" in (1 +x)", show that \"0 + "l/\^i + "2/(^2 + ^3/ + . . . . {f^n-i + "n) = (n+iy n ^'Oi **!> ^f a- ' • • • "« CHAPTER XXI. INTEREST. DISCOUNT AND ANNUITIES. for which interest is paid is called the Princioal • the i„f . on one dollar for one year is called the RatT' Id ^. '^"'' the principal and inte^st for an, gi.en tfnTe'rs'th: Amor„: 2. Interest is of two kinds, simple and compound. onW "r*. '"l"'^' " '"*""'''' ''"'""''^ "P"- th« original sum ^L f t .P"^"-' ol interest as it bTomes due Susur added to the principal, and interest for the succeeding p^ri«f interest. Ihe latter IS the only correct mpfK^ri f i'"""" interest which should evident,, ^r u^: *" he Xl Tb? J.tl.ut regard to the manner in which the' deU has tent Let P be the given sum in dollars, r the interest on one dollar Imou::. '^"' " '"" '""* '" ^"^■^- ' "■« '"'«-'. »d 'I' it rClVr '"^ ""•' ^^" '^ "■• ■""• "--^^ '- » y-rs i:^:-, (1) From = P{l+nr). (1) and (2) it is evident that if any three ^2^ of the (juanti- 316 HIGHER ALGEBRA. ties, P, /, n, r, A (excepting the three /*, /, vl) be given the other two may be found. 4. To find the •present value and discount of a given sum due in a given tivie, allowing simple interest. Let A be the given smn, r the given rate, n the number of years, P its present value, and D the discount required. Then Therefore And A = P{\+nr) P A i +Mr D = ^-P, ..A-/- 1 -{-nr Anr (3) 1 +?tr (4) In actual business, and for short periods of time, it is cus- tomary to deduct interest on the whole sum instead of the true discount which, as shown by (4), is the interest on the present worth. This is known as Bank Discount, which is therefore greater than true discount by the interest on the true discount. 5. To find the amount and the interest of a given sum in a given time at compound interest. Let P denote the principal, r the rate, n the time in years, / the interest, and A the amount. The amount at the end of a year is found by multiplying the principal at the beginning by 1 + r, and the amount at the end of each year is the principal at the beginning of the next year, therefore the series. P(l+r), P{l+ry, P{\+rf. gives the amount at the end of 1, 2, 3. . . ...P{l+ry n years respectively. Therefore And A=P{l+ry, I=A~P ;=P{(l+r)"-l} (5) (6) iiil (3) (4) (6) (6) . Interest, discjoUnT and AMNUiTifia ai7 ^. To find the present value and the dise AJfNUlTIES. 3id c^. The bank discount on a bill due in one year at 8 per cent, is $540 ; find the true discount. ^6. If the interest on $A for a year be equal to the discount on $B for the same time, find the rate of interest. 7. Divide $1,000 between three persons aged 18, 19, anTio years respectively, so that on their coming of age (21 years) their shares may be proportional to 4, 5, and 6, reckoning com! pound interest at 5 per cent. 8. In how many years will a given sum of money treble itself Tl) at simple interest, (2) at compound interest, 3| per cent • having given log 3 = .47712, log 1035 = 3.01494. ' C9. What sum of money at 6 per cent, compound interest will amount to $1,000 in twelve years; given log 106==2i253ifi log 49697 = 4.696329 ? 6 .w oowm, 10. In what time will $100 become $1,050 at 5 per cent compound interest; given log 2 = .301030, log 3 = .477121 W 7 = .845098? ^^. *og' n. A merchant's profits during each year are -, and his ex- 1 penses - of his capital at the beginning of the year. In how ™anyyea«L.will his capital be doubled ? (t^fXperson invests his money in a business which pays 4 per cent, per annum. Each year he spends a sum equal to twice the original income. In how many years will he be ruined; given log 2 = .3010300, log 13 = 1.1139434. 13. Find the interest on one cent for 2,000 years at 5 per cent. (1) at simple interest (2) at compound interest : Given lo- 105 = 2.0211893, log 23912 = 4.3786159. ii '^'*>0'iiii^^maimdi^iamm^mu»if.K m i 1 r mankn ALGfefenA. if n ANNUITIES. 10. An Annuity is a fixed sum of money payable at the end of equal intervals of time, usually one year each. 11. An annuity is said to be forborne when it is left unpaid tor any number of years. 12. A Deferred Annuity, or Reversion, is an annuity which does not begin until the end of a certain number of years When the annuity is deferred n years, it is said to begin after n years, but the payments being n.ade at the end of each period, the iirst payment will be made at the end of w + 1 years. 13. An annuity whi h is to continue for ever is called a Perpetuity. 14. A Freehold Estate consists of land, or other property, which yields a perpetual annuity usually called the Rent The annual income derived from freehold estates, irredeemable stocks, etc., IS frequently called a Year's Purchase. 15. Freehold estates are sometimes leased for a term of years for a certain sum ^n ca^h at the beginning of the period. Suppose an estate so leased, and that ;, years before the term expires the lessee wishes to obtain a new lease good for « + n years the sum which he must pay for this extension of time is called the Fine for renewing n years of the lease. lY/ 16. To^nd the amount of an annuity left unpaid fo- a given U number of years allowing (1) simple interest, (2) compound Let P be the annual payment, r the interest on one dollar for one year, A the amount, and n the number of years Since the payments are made at the end of each year, the first payment will be at interest n - 1 years, the second n - 2 ypara n.nrl an rvn TU. i ; -"-- '--". ^»us, wu nave, e at the t unpaid annuity )f years, fin after 1 period, called a roperty, Rent. ^eraable •f years period, le term >r p + n time is % given ipound lar for ir, the 1 «-2 tK'TtetlEST, blSCOtJNf AND ANNUITIES. 321 At simple interest from (2), ^='P{l+{n^.l)r}+P{\+{n-2)r}+....Pilj^r)+P = wP+(l+2 + 3+....w-l)7>r (10) „ n(n- 1) „ 4. I = nF+ — ^- — Lpj.^ ,y^T 7vwt.>v £<4^ ^ At compound interest from (5), = i'{l + (l+r) + (l+r)2+...,(l+^^„-,| Tlie reader should observe that the coefficients n-\ n-2 etc.. in the value of A at simple interest become ea^oLnts in the value at compound interest. Jr 17. Tojlnd the present value of an annuity to continue a given ? int7reZ. "^ '""' "'''"'"" ^'^ ""^'^ '''''''''' ^'^ --/---" With the notation of the preceding article, the values of the annuity at the end of the given time are, nP+-n(n-l)Pr and - {(1 +r)»- 1} Therefore its values at the beginning of the given time will be found by dividing these expression by 1 +nr and (1 +r)- re- spectively (Arts. 4 and 6). ' Thus, nP + -n(n-l)Pr I +nr and^|l--_i_l r [ (1+r)"/ s'Lttely"'""' "'"" '" ""P^^ ^"' ^^°^P^""^ ^"^^-* - Cor. The value of a perpetuity may be found from the above }^. i 322 HIGHER ALGEBRA. by making n infinitely great. The expression for the value at simple interest may be written i n , and when n oc the numerator becomes infinitely great, but the denominator remains finite, thus making the value of the present worth in- finitely great. This is another indication of the impractical results of simple interest when long periods of time are involved. In the expression for the value at compound interest. ^ '(l+r)* becomes indefinitely small when n becomes indefinitely great, p and the present worth reduces to -, which is evidently correct, for this amount of cash will give a yearly interest of P for all time to come. ^s/ / 11. To find the present value of a deferred annuity to commence p^t the end ofp years, and to continue n years, allotving compound J interest. The value at the beginning of the period of n years is r\ (1 + rf] (Art. 17) Therefore the value of this sum p years preceding this date is r(l T;^'(^-(lTr7} (^^-6) This formula also gives the fine to be paid for renewing w years of a lease, p years before the first lease expires. 0»r. The present value of a deferred perpetuity to commence p after » vears is . ^' r(l+ry Ex.—X. mortgage for $5000 with interest at 6 per cent, per annum is drawn January 1st, 1890, payable in 8 years; find its INTEREST, DISCOUNT AnC ANNUITIES. 3^3 cash value on March 1st, 1890, allowing the purchaser 10 per cent per annum, payable half-yearly. The annual payment of interest is $300. The values of these payments at the end of the period of time with 10 percent interest, payable half-yearly, are 300 (1.05)", 300 (1.05)'2 ••#. Thus the total amount will be .5Mf + 3#| {(1.05)" + (1.05y2-H n »5Mi + 3Ma^-^?>:izil 1(1.05)2-1/ = 8462.f6 We must now find the cash value of this sum payable in 71 years, allowing 10 per cent, interest, payable half-yearly. Thus, 8462.06 (Art. 6) = $3940.13 (1.05)i'*» is th«j cash value required. 19. The preceding propositions are sufficient to solve the practical problems arising from loans, stocks, debentures, and all transactions involving periodical payments for a fixed num- ber of years. Such are called Annuities Certain, but another and most extensive department of the subject relates to Life Annuities, which are payable during the life of a specified person or the survivor of a number of persons. For further mformauon the student may consult the article "An- nuities," in the "Encyclopedia Britannica," and "System and lables of Life Insurance," by Levi W. Meech, Norwich, Conn. EXERCISE XXXIV. ^, \nJ^^^ ^^^'^^ P*^""*^"* ^^^ ^^'^^ y^^ ^^" ^m a loan of $1,000, money being worth 6 per cent. ? ^ 2. Find the amount of an annuity of $100 in 20 years, allow- ing comnound intersRt a*-. 4 1 --'— .-— ■2 t^ I (."CIIU I given log 1.045 = .0191163, log 24.117 = 1.3823260. ^u htOHElk ALGEBRA. I 3. A freehold estate which rents for $216 per annum is sold for $4800 ; find the rate of interest. t 4. How many vpjai^R' piirny|g,^fl RVinnU bo given for a freehold estate, money being worth 3| per cent. ? 5. Find the cash value of an annuity of $250 to continue for 20 years, money being worth 6 per cent. ; given (1.05)'» = 2.653297. 6. If a perpetual annuity is worth 25 years' purchase, find the amount of an annuity of $625 at the end of 5 yeai-s. 7. What annuity, to continue 20 years, can be purchased for $10,000, allowing compound interest at 5 per cent? j^8. For what sum might an annuity of $400 a year, for 10 years, to commence in 5 years, be purchased, allowing compound interest at 6 per cent. 1 9. A person who enjoyed a perpetuity of $1000) per annum provided in his will that, after his decease it should descend to his son for 1 years, to his daughter lor the following 20 years, and to a benevolent society for ever after. "What was the cash value of each bequest at the time of his decease, allowing com- pound interest at 6 per cent. ? Given (1.06)-» = 3.20713547. 10. If 25 years' purchase must be paid for an annuity to continue n years, and 30 years' purchase for an annuity to con- tinue 2n years, find the rate per cent. 11. A man has a capital of $20000, for which he receives interest at 5 per cent. ; if he spends $1800 every year, show that he will be ruined before the end of the seventeenth year ; having given log 2 = .3010300, log 3 = .4771213, log 7 =.8450980. 12. A young man enters upon a situation at a salary of $100 per quarter, which is increased $10 e\ery payment. His ex- penses are $75 for the first quarter, and increase 5 per cent, each succeeding quarter. He invests his savings at 6 per er cent/ 4=/ 1,1 V' I \ m is sold freehold tinue for )mpound r annum scend to 10 years, the cash ing com- nuity to f to con- receives low that ; having 30. of 1100 His ex- er cent. )er cent^ INTEREST, DISCOUNT AND ANNUITIES. 325 per annum. What viH he be worth in 10 years? Proceeding in the same way would he ever be ruined 1 13. A merchant invests $12,000 which yields 25 per cent profit annually. At the end of the first year he withdraws $1000 for expenses, and each succeeding year 33 J per cent, more than the preceding year. How many years before he will become bankrupt 1 ^i< 1 4. The annual rent of an estate is £500 ; if it is let on a lease of 20 years, calculate the fine to be paid to renew the ) lease when seven years have elapsed, allowing compound interest at 6 per cent. ; having given log 106 = 2.0253059, log 4.688385 = .6710233, log 3.118042 = .4938820. 15. Find the present worth of a perpetual annuity of $10 at the end of the first year, $20 at the end of the second, and so on increasing $10 each year ; compound interest at 5 per cent. per annum. ^16. An annuity is payable for a terra of 2n years ; show that its present worth for the first n years is (1 +»•)» times its present worth for the second n years. If t' o present worth for the whole time is m times the present worth for the last n years find n. * (^17. A loan is repaid by an annual payment for n years of — of the given sum ; show that (1 -I- r)" (1 - mr) = 1. ^18. If there be n annuities of 1, 2, 3 .... n pounds respectively left unpaid for n years, find the sum of their amounts at simple interest. 19. If P be the present value of an annuity to continue for n years, and P+Q ita value for 2n years, find the yearly value of the annuity. , / on T?- J XI 0^^y^^AAV + AV. 18. If az" + 6y» + c«» be divisible by pofx' + vz) /'«2„u.«> \ then bp^- + cq^n ^ „^„^„ ^Q y ^^^ + y«^ - (S- y +i?'«)^. 19. Simplify 3 .5 and the velocxtiea of the water through the pipes a^ as 3 : 4. tt! Z\u '\''" ''''^ ^^^^'^^ "^«^« ^-« «-ed through the second than through the first. Find the number of gallons which flow through each pipe per hour. ^ 21 A man walking upon the railway track has partly crossed a bndge whose length is /, when he perceives a train app^rh Lg sulth ;-f •'' "'^"" ^ '^^"^ '''' ^"^^«- H- position i! such that ,t 13 equally safe to advance or to retreat. Find the train *'^'"' ^'''^''*'"^ ^' ^^' J"'' ^^'^^ *^ ^«^^P« *he tontermr^^'^^^^^^'''^^^^^-'^'^^^^^^^^^^)^---- 23. Sum (.-,). (^_^,(y.^ + . . . . to »i terms. 1). 24. Sum to n terms the series whose w'" terms are 25. Compare the length of the sides a, i, c of a right-angled triangle, c being the hypotenuse, when the squares described upon them are in harmonical progression. 26. If y be the harmonical tnAnn Vw^fwi^an -,. a^A - „- i j , , " '■^'" •" »"« >ii aim X and » be the arithmetical and geometrical means respectively between a and b, express ij in terms of a and b. mm 330 HIGHER ALGEBRA. 27. Show that a series oi numbers in A. P. may be found whose sum to n terms is always an even square, but that no such series can always be equal to an odd square. 28. Six men and their wives stand in a row. In how many ways may they be arranged, providing each man must stand be- side his wife 1 In how many ways, providing no man may stand beside his wife 1 29. The driver of a four-horse coach can choose his horses from a stable of 6 white and 8 black horses, but he must not pair 2 horses of different colore. In how many ways may he choose his 4 horses 1 . 30. Find the quotient when the sum of all the numbers which can be formed with n significant digits is divided by |n-l times the sum of the digits. 31. In a basket are 10 apples at 3 for a cent, and 5 pears at 2 for a cent; a boy has 6 cents in his pocket and wants some fruit; how many choices has he 1 32. In how many ways may 6 persons each choose a right and a left-hand glove from 6 pair without any person taking mates 1 33. If Pr denote the number of permutations of n things, r at a time, then n(r.-l)(P„_, -Pn-.)(Pn-. -Pn-z) • • • (i'. " A) = A • A • A • ■ • ^-2^. 34. Prove that 2"=! (n + 1 )n (n-Hl)n(n-l)(n-2) Li Li "t" • • • • 35. Prove that ^'6-'0-^ 23.3 1.3.5 2«.32 2 5* Li 1.3.5.7.9 2\3» . Li 1 , 23 2 ,- 1 36. Prove that 34" 3 ^ ^ ==2^" 2* ' 4 ""~2« [5 3 1.3.5 f ^7. In the expansion of (1 - «) " prove that the sum ot the co- APPENDIX. 831 efficients of the first r terms bears t^ ♦»,««:. the ratio of 1 + n(r - 1)^1 ^'o^ffie.ent of the r'*" term 38. Find the (r + !)«• terms of (1 - 4x)-i and — L-_ 39. Find the coefficient of x-* in the expansion of ^^^, 40. Sum to n terms, ^ ~ -^ ) 41. Show that the coefficient of w- in H - r)- i« « i . . «um of all the preceding coefficients. ^ ^ ^ ''^""^ *" '^' 42.- In the expansion of (^tfj^He coefficients of the (2.-1). and the (2r)«' terms are equal. and ^0<7n+m„.x+/>.^„_, + ....^0. 44. If ar be small, find the value of L i + 'v^i-ar r+Trrj* 46. Solve ^-^!±5!±£'^i^!+il+f^ 46. Solve iiJy,y + 2^ ^ + 2;g 3» 3a: 3^ = ^ + y + 2. 47. The numerator and the denoDiinator of one frartinn each grea^r b, 2 than the com^ponding term's oTa l^ZZ w iir 332 HIGHER ALGEBRA. of each were increased 1 yard, the former would make only 50 revolutions more than the latter in going the same distance. Find the circumference of each wheel. 49. Solve x + y + z = ly xy + yz-zx=l, (he a\ 50. Eliminate x, y, z from x = yz\- + ---^j, (0 a h\ ^ \z X yj la h c\ 51. Solve V^^= ^V^\Z + ^ ~ E) ' ^9 Tf — = ^ where d is the diflference of the roots of a;^ + mo: + n = and i> the difference of the roots of x"" ^-Mx+N= 0, then m^ n M' N' 53. The coefficient of x^ in (1 - ax)-\\ - bx)'^ is 54. Show that the equation, x- + rx-^ + s = will have equal roots if {-(i>-n)V={^(n-;^)| 55. If the roots of x"^ + px + q = are real, so also are the roots of the equation, {mp + 2)x'^ + 2{mq + 6p)x + 18g = 0. APPENDIX. 333 56. Solve ar* 4- 1 = 2(1 + xf. 1 57. ITove ^- j _ -|---____| if^ 18 nearly equal to q. 58. If r < 1 and positive, and w is a positive integer, show that (2w + l)r^(l - r) < 1 - r^+\ 59. Show that the sam of the products of the first n natural numbers, three together, is {n-2){n-\)n\n + \f 48 ■ • 60. Find the condition that the equations, Ix^ + m'if + ns? = 0, ax + hy + cz = Q, may have only one set of values for the ratios x-.y.z, and show that if this condition hold, Ix my nz a b c ' 61. Sum to n terms and to infinity, J_ J_ 1 1 1.6'^6.1l"*"ll.l6"^16.2l'^"** 62. The equations, a^ + ^ + i^-. Sxyz = a% yz + zx + xy=> i', x + y + z = c, cannot be simultaneously true unless c^-a^ = 3cb^; and if this holds, they are true for an infinite number of finite values of X. y, z. 63. Show that (l+a: + a^)(l+a:3 + a:«)...(l+ar'"-' + ar'»»"-') = l+a: + ;r'^-H...a:3"-i. 64. Prove that the coefficient of ar^^ in the expansion of (1 -;r)(l+ar)^ IS (n+l)(4n2+lln + 6) 334 HIGHER ALGEBRA. I ! 65. If ar be a positive integer, prove that — — — ~- is a positive integer. {l-xf 66. If mx^+ny^ = a\ mx^ + ny^ == a\ and mXiX^ + ny^.i = 0, ft yv2 then x^ + x^ = — and yi-\-y^ = m ' '- n 67. If nPr denote the number of permutations of »i things taken r together, and I{,P) denote „A + „A + • ■ • • n^„, show that IU,P) = (n+l){I{,P) + l}. 68. The coefficient of x"" in the expansion of (l+a;){l+cx){l+c^x)...., the number of factors being unlimited and c less than unity, is equal to C»r(r-1» (l-c)(l-c'0(l-c=')....(l_c'-)- 69. There are p + q numbers, a, §, y , of which p are even and q odd. Show that the sum of the products, taken 3 and 3 together, of the quantities, ( - if, ( - 1)^, ( _ 1)^ . . . . , etc. '=Q{(Q-py-Hq'-p'') + 2(q-p)}. 70. If and A=aQ + aiX + a^x^ i a.^x" -5 = cTo + «iy + «22/^ + . . . . «„2/", n n show that when ao = ««, «i = ofn-i, etc., A'.B = :^:y\ where x and y are the roots of x"^ +px +1=0. 71. li X, y, z he three positive quantities whose sum is unity, then will (1 - ar)(l - y)(l - z)>8xyz. >, 72. Ifa; + y + « = «2 4-2/'' + 22^2, then will x{l-xy==y{l-yy = z(l-zy. 73. The equation, {x+ V^^Tc){y+ Vf-ca)(z+ Vz^-ab) = ahc, is equivalent to ax'^ + hy^ -\- cz^ = ahc + 2xyz. APPENDIX. 335 74. Prove that the equations, a b c ^ y ^ ^ -3 + T3 + -3 = 0, a-* b^ c^ * are equivalent to only two independent equations if bc + ca + ab = 0. 75. If a, § be the roots of a;* + a; + 1 = 0, show that -_i-_^ W^ L.\ l+x + .r' a-(i\l-ax l-/3arj* 76. If a, b, c be the roots of the equation, ar3 + ^ar + r = 0, form the equation whose roots are ab a +b 1 A 1 . 1 , oc + j—~ and ca + b+c c + a 77. There are n lines in a plane, no two of which are parallel and no three pass through the same point. Their points of inter- section are joined. Show that the number of fresh lines intro- duced is ^(n)(n-l){n-2){n-3). 78. The number of ways in which r things may be distributed among n +p persons so that certain n of those persons may have one at least is {n+py-n{n+p-iy + -^--^(n+p-2y- li 79. If the quantities r, y, z be all integral, and satisfy the equations, y^ zx ~ ~^y • each member of the equations =a^ -\. and xyz{xy+yz + xy) = x + y + z. 336 HIGHfift ALGEBRA. r,^ Tm aA-n(a-b) a-bx ... . , , ., « ., 80. If x=- )■ rf, :-; T-„ will be equal to the sum of the b + n{a-hy (1 -xf ^ first n terms of its expansion in ascending powers of ar; a, b being unequal quantities. 81. The number of combinations of 2n things, taken n to- gether, when n of the things, and no more, are alike, is 2"; and the number of combinations of 3w things, n together, when n of the things, and no more, are alike, is \2n 22n-l 2( [nf 82. The number of ways in which p things may be distributed among q persons, so that every one may have one at least, is Q'-9{9-^y + Li ^(^-2)^-.... 83. Prove that = 0. n(l+x) n(n-l) l+2a; n(rt-l)(w-2) l+3a-' 1+nx ■*■ [TlT+nxf [£ {l+nxf'^' 84. Factor (a" + 6^ + c^)xyz + {b^c + the equation, ay-bx = cV'uZaf:^Tr-rg^ unless c2 is less than a2 + i2. / \^ / . 101 Write down all the numbers that can be composed of the four digits, 3, 4, 5, 6, which are divisible by 11. 102. Six papers are to be set in an examination, two of them m Mathematics. In how many different orders may the papers be given, provided only th.t the two mathen,atical papers do not come together ? 103. Find the sum of the series, _L _?_ 16__ 25 1.5 5.14 14.30 '''30~.~5r)'*' *«»* terms, the^ last factor in the denominator being tlie sum of the other i.actor and the numerator. APPENDIX. 339 104. If n ia greater than 1 in the series, 12 3 4 n n- w n* n show that the sum to infinity is -— „ and the sum to m terms, {n—\y ' w(n'^-l)~m(n-l) 105. If x — a y — h % - c b c a X — h y — c z — a = 0, + ' cab X — c y -a z — h = 0, a + = 0, show that x = yT=z = b e ab + bc + ca mc au XI . (*-c)(* + cf + anal. + anal. „, ^^'- Show that |,_,;;,^,;.^,,,,^ -^^=2(a+^-,c). 107. Show that n-" = (-l -?!i.-J ^ (2n- l)(3 n-2) ^ |»- Ul Li . Li '"} 108. If rt, />, c are the roots of .r^ -px^ + qx -r = 0, express 2a^^ + 2b^c^ + 2 c^a^ -a*-b*-c^ 2ab + 2bc + 2ca - «« _ //i _ ^2 in terms of 7>, 5- and r. 109. If a-(a;-l)2 + 2/(y-l)2 + s(«-l)2-y«(a^+l)2-2.r(2/+l)* '-xy{z + lf + ixyz = 0, and a: + 2/ + s>l, prove that (ar + l)(2/+l)(2! + l) = .T2 + 2/2 + ;52+l. 110. If X* +px + 5- be divisible by x'^ + ax + b, then will (a^ + jo)(a=' - /)) = 4«'^5' and (/>« + ^){/>2 - q) ^pV,\ d40 HIGHER ALGEBRA. I 111. lif{x) be divided by ar-a, and the integral quotient by x-b, and the second quotient by ar - c, the remainder will be _ {h - c)f{a) + {c - a)f{h) + (g - h\f(c) {b - c){c - a){a - b) a + cx ^^^' ^^ c + ^ ^ expanded in series ascending by powers of (l-x) and (1 +x), and A and B be the coefficients of (1 -a-)" and (1 + ar)" respectively, then U-^rC^) 113. On a raUway there are 20 stations. . Find the number of tickets required in order that a person may travel from any one station to any other. 114. Show that if a" :< 1, a must be ,=,<, 1. 1 8. -7- or oc. 10. No definite value. 11. ±^ -2(a - h)' EXERCISE II. (Page 23.) 1. 25:49; 17:20. 2. 10: 11; ar+ 1 : a; + 2; a' + i^ja' + iV 3. 3:5, ratio of equality. 6.(l)i±-^; (2)3or-^ (3) 2 or -J r (4) "("+•> (t - c 6. 35 and 65. ad — be 4. 495 and 693 1 2 c-d. ' npt mqt 9. 17. r mq — np mq — np 21. mq: np. 7. 82.40. 15. The latter. 19. 56, 7:8. 22. n^qr-.m^ps. in(a — 1)* 8. 929, 260. tiq — mp 16. inq + np + Imp' 20. 2:3 and 4 : 5. 23. 6 min., 4 min. 24. qv'^ps 25. q + r-p:p-{-q-r. EXERCISE III. (Pagb 31.) 1. 1:-2:1. 3. x= -y = « = ±2A/2l. 2. ha+a):ha+c):\-ac, 4. a:=10, y=16, !S = 7. 342 HIGHER ALGEBRA. 10. a;«=a-6, y = 6-c, 2 = c-cf. 11. ar = 6 + 2c, y = c + 2a, « = a + 26. b-c 12. ar=.64-c-«, z = a + b-c. 15. x = 5, « = 3. 13. a; = 6, U. A = a ' c — a 16. ar5 = 6V 6 ' a — b 2^= a ' z = 7^ = 18. a6c + ygh - ap - bg^ - ch^ = 0. 21. 2a6c + a6 + 6c + ca = l. 6 ' 19. a3+43 + c3-3a6c = 0. EXERCISE IV. (Page 41.) 1- 25. 2. 8. 3. 7 + 5a/¥, i/y+ l/5. 4. 7. r 5 7or-19. 6. m^^,m^^. 7. 3, 12, 48, or 13|, 22J, 37^. 8. 9 or 11. 12. -^- . 13. ^, ^ . 16, ^^""^ 24. $40 or $60; $50. 25. 30, 45. 26. 7, 8, 9. 27. 3J. 28. (//t + rt)(jr-j9s):(jt? + g')(7na-wr). 31.64,36. 32. (2a - b)c (2b-a)G 3ac 3bc a + b ' a+T~' a+'b' a + b' EXERCISE V. (Page 54.) 1. 45. 5. 10 or 15. 2- 10. 3. 10:1. 6. ^. 7. -. 5 a 9. 6, 18:25. 11. n?a. 12. ±16. 23. 4 : 5. 24. 244^ ft. -000906 in., nearly. a -2b + r 26. dollars. 26. men'' (w + lK 4. 2:3. 8. .= ^«l 22. 143. IT (m + l)b^ ANSWERS AND RESULTS. 343 27. 6.376 cwt. 29 224J days, nearly. 28. $93.75. 30. 1 day 18 hrs. 28 min. EXERCISE VI. (Page 63.) 1. 23,33,203. 2. 14, -7,38-3n. 3. (l)3n + l; (2) 11 - 3n; (3) IJ, 0; (4)37,5^-33. 4. 34"', 46'^ no term. 5. 7, 290. 6. 194. 7. {n+l)a + (n-i)b, 2na + {2n-5)b, {3p + iy\ 8. (1) 5050; (2) -790; (3) -333; (4) n*; (5) 1325 V^ 3; (6) -n. 9. 25, 16,m. 10. 5,8,11,14,17. 11. x,2.t-1...x^-x+1. 12. -(ni' + mn + n'). 13. 7,4. 14. 8, 8w-14. 15. 59. 18. 5^ ir-^^' 19. c. 21. 7. 23. 3, 6, 7. 24. 20. 25. n'. 27. ±l,±3,±5;6jpj2^4j?. pq+\ 16. wl 17. 110. 22. 6, 11, 16. 26. 8, 10, 12, 14. 12 3 4 98 ^ 1 1 _ " * 10' 10' 10' 10 29. 34.^,(2-^), n-\ ' 35. 30. 3 or 8 hours. (jt> - 2)(2a - d) d{i-p) • 36 !!i!Lzi]+i ^(^ + 1) ^(^'+ 1) n(yt+l)(7t' + n + 2) 2 ' 2 ' 2 ' 8 • 38. n^n' + l) EXERCISE VII. (Page 72.) 1. 11 or 15. 2. 2, »t(n-2) (2w + V)(ma — nb) 3. (13-n)(2n+l) -^ \ 5. 91, r^-r+l, w2 + n+l. 6. lOw-8, 10. 7. 25 or - — . The sum of 76 terms of 0, - , - .... is 950. «* 3 3 S. 2f. 9. 9or-ii. The pum of 11 terms of 19, 17. ... is 99; the sum of 11 terms of 1, - 1, - 3 is - 99. J 344 HIOHER ALGEBRA. 2a 10 -7r 11. -T is a negative integer. 13. 3:2, -4:5, . d 3r — 1 ._ m + n (2q-p)m+pn ^^- -2~' 2q • 21. '^(^. 4 16. g, 2, 3.... 20. (n-l)«. 25. (n -q)P-{n-p )Q p-q 30. 1 :ai + 6c + ca:a + 6 + c. 35. 1,3,5,7.... 39. n-'-n + l. 40. H^^-J-^D j' Hn^--!) !' 1. 32. 5. 22". 2. 19683. 6. aV"-'. EXERCISE VIII. (Page 80.) 1 3. 7. - 256' J2»-3 a ,2n-3' 9. -^(v'3-1)". 10. {V2-\f l..M..^{(|)"-l}. 11 1?23 1 1 1024 •^"2'»- 15. 4. 2(3)''-». 8. (2n-l)a"-V'-'. 11. 1023, 2--1. 1 _ 2*'» 1 + 2'»+i ~l 3 ' «'{1- (-aa;)"} a« {J^+ (aar)*-^} a:(i + aar) ' ^ 2»+i 13. a;(l+aar) 16 (2+V^3)n.f./3 -2 ^^ a{y^-\) ^3 + 1 ' ' r\r-\y ' a»+'(a-62^ 5-n n 2*(2*-l) 15 -GM- 20. 23. 3, 3(2)-^ 24. 5^ 25. \, 2, 12. 26. 1, 3, 9. 98. 12, i 6. 27. -{m-\. Vm?^^v?}, -{m- Vni'-in''] 4 2 9. 3, etc. 19 19 i. i, 2. etc., or—, -— , efiQ, 1. /. T- ANSWERS AND RESULTS. 345 33. 1. ±1 1? ±5f 17' ^17' 17' ^17* 32. 2^, 5, etc., or - 7J, 15, etc. 34. 2,4,8,12.16.oriL6, _?l?,etc. 35. 5, 9. 13, or 23, 9, - 6. 36. 5. 10, or - 8i - 323i 37. 2-2(2- + 2-» - 1), 2"-»(2- - 1). 3 J 38. 2»+»-3, 3.22"-2-2», (2- -1)2. "(ii-i) »(n+l) 39. 2 ' (2» - 1), 2--^ _ 1. 40. 22"-i _ l, 1 12^" - 2'"-^ 42. a2(l-r)(l-r'") 1. 1. 1 +r ^■l- 43. -i*'^" 2a Vb Va+ VV Va+ VT 7.#-l. 8. v'2+1 11. 4 + 31/2". BXBROISB IX. (Paor 87.) 3 1 '2" 2 - 21 5 + SV3 „ 3(3 1/ 2 - 2) 1 1 « 8 2* ^- 21' ^- 2(2+ V2). 9. 10. aVab 12. 3+ v'3 13. 1^ • ■ V^b-i n.r" ar" - 1 1 or-l (x - 1)2' (l-a;)*- 14. 4-(n + 2)2-»+>, 4. 15. 6-^ti 6 16 ^ . 6n+l 2 2»-i ' • • 9-^9(_2)'-i' 9' 19. c2»+i. 17 Vr») q(l-&'»)r"+^ (l-r)(l-6r) (l-r)(l-6)' (l-.r)(l-6r)- 20. (a6)"»'. 21. (mn)*, rJl^y". 22 /'^'\^' 30. \/6 l/3- 'v/2 31. v/2' 33. a = !!^tL) .= 1 32.i^(10--l)-^. 34. 4n w+2 ' ■ w + r (r-l)2 r-V (r-l)(rP-l)~;ni- «** ,, /a/"5+1\ ^^' (—2—)^' ^' a-" <*-''' 2(«2-62)*" 346 HIGHER ALGEBRA. EXERCISE X. (Paob 07.) 1 -1? 5 64 32 64 7' T' l3- 7-n 3. 1. 4. 17, ±8,31?. 5. ?,? 4 7 1 11 4'"^' ~4'632;t- 8 ah {n-\)a-{n-2)b' 11. 3f. 9. a + h. 10. 3, 1. 12. 14 or -. 13. 20J, 4. 2' ' 2' 4' 6' 8' 10' ^'^ 24' 17' 2' 3' ~ 4' ~11' " l8* 15. 104, 234. 16. Half the middle term. 17. 2, 3, 6. 18. a,6,c. 24. )^_^ ' . 27. bc{q~r) + ca(r-p) + ab(p-q) = 0. 2mn 2mnq 28. m + n* 2nq ■+ (m - n)p' 34. No. EXERCISE XI. (Paob 107.) 1. 2870. 5. 4466. 9. 1375. 15. 120. 19. 1981, 25. 2. 4960. 6. 5915. 12. 330. 16. 50. 3. 3920. 7. 4944. 13. 4970. 17. 20540. EXERCISE XII. (Page 109.) 1. 2w(n+l)(2w+l) 4. 6214. 8. 220, 385. 14. 20, 15. 18. 2024, 3795. n(in^-l) ^ n{n-l)(2n-l) , 4. n\2n^-\). w(?i+l)(n + 2)(3n + l) 7. (-1) i»+i 12 m(w+ 1) 6. n(n+l)(w+2) 6 8- 9i(^^*'-i)(-ir'-i}. 6 6 5' 4 ANSWERS AND RESULTS. 347 9. n(2n'-3n-l). 10. ^(^+l)(^ + 2) np(n+l) 13. 8,1,5 or --,.!,__. 11 -15!^i) "• «• #.• 16. ^(^^-l)(^+ l)(3n + 2) 24 16. 2»+l, 2"+i + 2, etc. y^(w+l)( 2n^+2n-l) 20, 2 21. - 22. !^l)|6a4-(4n-l)rf}. 12 23. 8{(-l)"+»(4n3^6n2-.l)_l}. 1 25. -n(8n + 9), j{7 + (- l)«+t(8n2+ i8n + 7)}. 33. 9, 15, 25. n 39. w+ 1 w(2w+l)(a-&)a 6(n+l)6 46. r2(- + 4)(2-+l),^(2n»H-n + 3), ^{n+ 1 +(-l)n(,_i)j, 47. 2(n^ + w), ^ ^+l)(^+ j). 2^ n(w+l)(n + 2)^ 49. w i''(M+l)' {(-ir+lK + 2n, !iV + a'&'-aV 26V6^r5 ' 16..= + ^*- J^-^ '/a«+A«+c«-3a26V' A* - c»a' y = 0: ^ lab 1 A/a«+66+c«-3a26V' C* - tt«62 l/a«+6«+c«-3a«6V" 17. x^-{±Vh^+2d^±V^a^-b^)}, '-{± ^^r^2±^2^+-p}; y=3{±V^A5^T2^TA/3(^C62)|^ l|j. v'6r:72^V2^rj:^j. 2 = ± ^2a'« + 62 ^{T2i/62_«2^y'2a2+6''}. 18. a; = ±4, i^v^S; y = ±3, ± 1a/3; «=±2, t| V3. 19. a;= -2, 2; y= 4, -1; «=-l, 4. J 356 KlOfiEft ALQEBnA. 20. x=i, 4; y=l, 3; «=6, -2. 22. «= 6, «= 1. 21, oi?=-r— — — -— — i-— etc. (See Art. 207, Ejt. 5.) 23. a: = a, 0,0, (ca-i2)(„A_c») y=± 2A/3a6c-a3-68-c'' (c + a)'-(& + c)(a + ft) 2 A/Saic-a'-A^.c' ' _^ {a + bf-{c + a){h + c) ^ 25. ar = T~r{{^^ -ca + ah){hc -Vca- aJ)}, etc. 26. x{ac -ah- he) = y{ab + ac- he) = z(ab + hc + ea). 27. x=^(2a-b-c), 3ttAc-a3-i3-c'* y = 0, h, 0, etc.; 2 = 0, 0, c, etc. y=-(2i-c-a), z=-{2c-a-b). 28. a:=lj, 2§; 2/=2§, 1|; «=2, 2. 29. cc y= _ f(a»6»-c«)(a»+^)'» J (c=» + a7 31. aj= 0, ^ ; 1 + V^ :f V6"- 1 30. a; = 3, 4, 3; y = 3, 3, 4; a = 4, 3, 3. y= » = ■ 2 v^-1 2 2. 32. x==l, 9; 33. a;^6, 3J; y=2,-6; y=^6J; « = 4. « = 3, 4J. 18. 21. ANSWERS AND RESULTS. 357 BXBBOIaE XX. (Paok 308.) 1. ^+263-3a'. = 0. 2. a^- 7aV+ 14aV-7a.« = 6^ 3. a' + b' + (^ + abc=>0. 4. (? + 2/\ ^ g. 7. (b + c)^ + (b-c)^ = 2a. K 1 1 1 a' 6» (a -by 9. 1 1 1 bc-a' "^ ;i;ip + ^6ip = 0. 10. p\a + 6 + c)^ = 3. 1 1 1 1 1. 11. a?+b'^ + c'-ab-bc-ca = 0. 13 ^ . . a+1 6+1 c+1 rf+1 14. {x-h){y-k) = Q. 15. 2,^0 or x = 0. 16. y'-8 = or a:»-8 = 0. 17. 2y^+l=0, y»-4y + 3 = 0. «• ^63-8 «^ ~6^~ + a»=8- 19. {ac^-a^cy={b^c-hc,){aj,-ab,). 20. 4«*68- 2aV-a86*-6» = 0. ^^' b*=^;r^^- 22. p' = 6«(a» + ;n«). 23. c' + a»-6^ = 0. 24. im^ca + in^ab+p^a^ = imnap + ia^c. 25. (ail - «i6)' = (61C - bciY + (ac, - a,c)\ 27. By squaring the equations and adding we get (a« + i«) + 2(a* + 6> + (a2 + i2)^ = y2 + ^^ (l+ar)^ -yg-g2 But a2 + 62=l, .-. a262^ 4a; + 3 (1) Also, by adding and subtracting the equations and multiply- ing together, il^.y-a^b^J^^, (2) From (1) and (2) a' and i' can be found, and their values substituted in a? + b'^=\, 32. "'^ — ^.gjyz ^)- ?/' 28. ^. + ^, + i = w n' y{x~z)~i/ 358 HIOHISR ALGEBRA. 33. a»-3a62 + 20^ = 0. 35. a + i + c = 0. 37. a' + i''4-c' + 2«6c = l. 39. {a + b + c-iy = iabc. 34. a'-3aft»+2c3-6rf' = 0. 36. c' = a-^a + 6 + 2). 38. o' + 63^c=' + a6c = 0. 40. a=' + 63 + c-''-3a6c = (/'. 4^2 orx .*.. 2 41. aV + *V + cV = ^rTV2 + f'% c%^ 42. a6 + 6c + ca + 2abc =1. 43. (a^ ^ b'i j. 0^)3 + 8(a6 + be + ca)^ = 0. 44. .i;3+2/2=s5 45. 6(«3-i3)(2a»+63) = 9a(a5-c«). 46. (f!±/)*-(f!i-^y=i. 48. {a + bf-{a-bf={%c)^. 47. a6 = c + l. 3 7 9. Between - and - -. BXBROISB XXI. (Page 217.) 1. (x-2y-2a)(2.r-y+3a). 2. {ac - 2df = {a" - ib){c'' - U), 17. Between 2 and - 12. 21. w=-2. 22. m=±7. 24. (/wi - /in)'' = (Im^ — /im)(wni - wiiw). 26. (aai - iftj)" + 4:{ha^ + hj)){hbi + Aja) = 0. 29. (oci - aic)2 = (ail - ai6)(6ci - 6ic). 31. 576. EXERCISE XXII. (Page 223.) 1. X — , 3, 2 3' 2. ;r=2±V3, - ^, - ^. 3. x=-\±V -\, -\±V% 4. x^±V\ l±2v/"^, -1. 5. (1) :r*-2.i:'+25 = 0, (2) x^- 8.^2 + 36 = 0, (3) x^-2x^+ 9 = 0, (4) a;*-10.r2+ 1=0, 6. «*-10x'-19a;2 + 480;r- 1392 = 0. 7. ;r«-l6x« + 88.1;*+ 192a;'' +144 = 0, ANfcJWEK8 AND RESULTO. o^r. BXBROISB XXIII. (Page 230.) • 4- -2, -j^. 6. (27r + 2/,»-9jt,y)2=4(;,'-3y)». 8. (1) ^+2;.,r=' + ar(;,2^.^^^^_^^^^ (2) 0:3 _ ^2(^2 _ 2pr) + a:(;?V - 2yr») - r* = 0. 9. ar»-2y:r2 + ?'o: + r»=0. 10. ^== -8r^ x=- .^±3^5 ' 2' ~I • 11. m' = 3n. 13. 45' = 4(m+l)+;,2. 15. (2x^ + x + 2y~5x\ 12. qc-g^a-pq(b-pa) = e, qd-pe-q\b~pa) = Q^ 14. {e^-CA){D^-AF)=.{BD-EAf 16. (l)4W=8arf' + er', (2) 63 + 8«V = 4aAc: :r= 1 1 _! Q^ o/„ . . ' 2' ^* 21. (1) 23. 0. % (2) ?i?^), (3) z»z£r>. p 24. 1. />- OQ 1 3 9 28. 2 + 4^+8^' + -.. 2"r-^r-!6^+--- 27. «=1, 6=ll,c=ll,rf=i,^»o. 30. x = a + 2c, y = 6 + 3c. 3j w(w+l)(w+2) * 3 ^• 32. |*ifctl>V. 33. n'. EXERCISE XXIV. (Pagb 239.) 2.25. 3.300,1190. 4.20,36. 1. 604. 8." 60,125. I L. 10 r'''^ ^^^^^«00, «^ 12. 720, ^1«*00 55440 13. 34650. 151200, 121080960. 14. 114. 15. 300(^16. 60, 12. 17. 35. 30240, 19. 90720 7nRR7r-iT7^?r «« .,„ [20 90720, 70560, T76i(T. 20. ■v.. ■' L 13. i. 8 J 13 I 8. A 360 HIGHER ALGEBRA. 21. 122880. 22. 494020. 23. 27, 10, 18. 24. 80640. 25. 40320, 5040,-2520. 26. 24. 27. 3628800. 28. 282240. 29. 298598400, 8233505280. 30. 24, 24, 73. 31.362880,2903040. 32. \m. \ m-\ . 33.81126230400. EXERCISE XXV. (Pagk 250.) 1. 210, 84. 2. 38760, 3060, 8568. ' 3. 20. 4. 46558512, 5587021440. 5. 630. 6. 51. 7. 1023, 512. 8. 127, $762.24, 171. J. 791. 10. 24, 30 (including unity and the given number). 11. 163, 3393, 3386880. 12. 576, 821; 46866, 314695. 13. 36 or 39 cents. 14. 5, 2. 15. 8 or 9, 2 or 3, 243100. 122 16. jYj^, 222-1, (.•^-18)(25-1). 17. 2". 18. 12. 19. 8. 22. 390625. 25. 21, 56, 20. 12, 4. 21. I m I TO - 1 m-n 23. 5. [n + 5 5_[w,' 28. 8204716800. 24. (j9+l)(g' + l)2"-9-^-l. 26. 1820. 27. 56. [^ L n 29. TO -r I 8 — ■ • \r + s. — a I n — 8 30. 244. 33. 31. 19. 32. n{n — \) n{n-\){n—2) LT"' [I ' n{n - 1 ) n(n - 1 ){n - 2) n{n - 1 )(n - 2){7i^ - 1 3n + 20) [1 ' Li 48 n{n - l){n - 2){n - 2>) 8 • 34 n(n-l) p{p-\) _I^__. _LI^^ * LI LI Li \'^-p-^ \ 1 | w-;>-2 35. - I n ~ 1 . -f /?(;?- l)(w-;>) 10. 16. ANSWERS AND RESULTS. 352 36. !!^^!!i:il)l!*_tZ) . !i(^^) 0^+!^ ^^^ 2 "^ 2 "*" o 38. 6084, 37 204 M^tiK2ri+l) 6 (^ + limp. 1. 20, 10. \nr 4 ±-=^ 7. 1001. '26 125 10. l--=Jr=- ■ [5 1^20' EXERCISE XXVI. (Page 259.) 2. 5775, 34650. 3. |26 6. 1663200. 1100 • [95 ' ^^ • 199 11 -i=-- 11 [^- \r + n~ I \ri + r 13. -L:=:r- L__Z_ ?l I ?• - 1 H - 1 11. 7. 123 6. -1—. 9. 75600. 12. 9G9, 1771. 15. 2-". ifi 'i/^ A /" - ^y- ^"' 2 \2 ~ / *^*' I — 2~) ' ^^^oi'tliiig as u in en 17. (h+1)-. 21. 46376. 22 24. 690. 18. 30786. I pq + r ven or odd. 19. 576. 23. — 1'^=: ('7+ir(Li)^'' 25. 3^t- + 3w+l. 26. 209952. 27. 281.5 EXERCISE XXVII. (Pagk 265.) 9 1-- 10 M'lti)!!!*-*-!) 2"' ■ 3 • 11. n n -JfV IG. a= + 4(/c = />». 362 HIGHER ALGEBRA. 1. 2. 4. 5. 6. 7. EXERCISE XXVIII. (Page 275.) 3^ 4- x\a + b-c) + x{ab - be - ca) - abc. a;*-10r» + 7.r2+162jr-360. 3. .c* - 225.c2 + 1620.r- 2916. x5 + 5aa;* + 1 Qa^r^ + 1 Oa?x^ + 5a*.e + a^. 64a' + 576a56 + 21 60a<62 + 4320^353 +486o«26* + 291 6a6^+ 7296". 16a* - 32a'y + 24(/y - Say^ + y*. 1 - 1 2a: + 60j;2 _ 1 60^3 + 240i:* - 1 92j;5 + 64a«. « . TO 4 . en ■> icA 240 192 64 x^ + 12a;* + 60.r2 + 160 + -^ + — -^ + — . a;^ a;* a;« 4 1 a:« + 4ar* + 6 + -^ + -. 12. 15. 18. 09 1 25 1 40 l^=-a^b'\ 13. ...'—. 4«.r«a^<. 10. 252.cy. 11. -20000.r3. 10 14. 5 I 5 LiL 3^. 1 12 112 -^2V/. 16. --^-^.*:^ 17. (-1)- ■ Lr.__3r..p.-r r^ w ["_ ir n — t -.i;2'»-2y. il6 ^'- ^-^Vlt^-'^"^- n \{n + r) ;{n-r) 23. (-1)"- in \n In 24. n-7. 25. n = 7. 27. 26. 2j;* + 1 2a;y - 1 2a;2 + 2.v* - 4y2 + 2. 31. 16(4m*-l)Vm*-l. 29. 1. [2n 30. 192' 2n T 4 m (when n is a multiple of 3). T 32. L n (2n-r) 3»t + r 5 5 . 33. [2n n I 4ri + «j, [ZEI 6ri — m . 34. 2'* + n.2'*-». ANSWERS AND RESULTS. 363 1. 4. 7. 10. 13*'' term. 5"" term. EXERCISE XXIX. (Page 281.) 2. 5'" and 6"' terms. 3. 4"' term. 6. 540. 5 * ^' 3- 108864. 8. 512. 9. ,.= 14. x = 3, n.= 5, a = 2. 11. n = 2r + 8. 12. 2r = «. 1. 3. 4. 5. 7. 9. 10. 11. 12. 13. , 3 3 ' + r-32 x^-\ EXERCISE XXX. (Page 293.) 5 . 128 2 1 4 i+..-1,,hJ..'-.... 1 1 1 -i o lb 2^-2*a;-2~^3.c2-... l+6.r + 21/- + 56.r"'+ .... 1 ,, 4 20 , 320 , l-3.c2+6.r*-10,r" + .... 6. 1 -3.r + 6.i;2-10,r3^ _ o 1 3 3^5, ^•8+l6" + !6^ + 32"' + - «- - 2 a;3 1 .7;6 3 ' « ~ 9 a* , 3 27 ^-2" + ¥ ,k' - 135 2. 1.4....(3r-o) 3'^ a „3r ,3r-8 16 ■ r — 1 a; ./;-(ri + 1) 14. 17. 19. -x\ -63 "8 -225 _T «» a (« + iu-:;».+i )...^_{(r-_n,. ^^ [r. 15. r»y.. .-r' « nr+l + 16. 35 -X u 2" 16 e« ".c" 18 -Il(LiM-.5JHiiIl?!' 79 3^- ^9 364 HIGHER ALGEBRA. 21. i-ir.h±^i:^^^_. 22. 3 { j7 ( - 2r . 2/'--(r + l)(r+ 2)2-y. (r+l)(r + 2)(r + 3) il •1 -(r+i). nr 23. (-1) .1.3 (2r-l) dHri±l) 24. 25. 27. - 2"! X a 2/' 13.11.9.7.5.3.3.5.7 11.9.7.5.3 3'-'* X 21V. ) X .11 15 , 3.5.7.9.11 32' )10 L: 26. 28. XT ^7 3.5.9.13. 17.21 2^ (r+}y r + 2)(r + S)(r + 4) 29. x* + iar'+lOx-'+20x. 30. 1.3. 5. 7. 9. ...21 or .tr8(l +4.r-i-10.r' + ). ill 2'i x" 1. ThtU^hor5 3. The 39*'' term. 5. The 5"' term. Bl^ERCISE XXXI. (Page 297.) terms. 2. The 23"^ term. i. The 12">term. G. The 7''» term. 7. The 3'-'' term. 8. The 9'" term. 9. The 8'"^ term. EXERCISE XXXII. (Page 308.) ■^•-l^- 2. (-l)-i. 3.2m'+'Sm + 2. 6. (1) 9 99666...., (2) 10.00666 (3) 6-99927 + ...., (4) 5-00128. «-('>l-T'(^)3(l-^)- 15. -245 16. Coefficient of ."• is 3>. 2-3'-=. u'^- coefficient of .r^^+i is _33r+I 2-3r-.,,,-3r-3^ aild of .r''-+2 i, 0. 18 0:+ 1 )(^+ 2)(r + 3)(r+ 4)(r + .5) 30 55. mn+ -m(m+ i). • 23. 3V3-2. 53. 462. ANSWERS AND RESULTS. 365 EXERCISE XXXIII. (Pagi 318.) 1. $1080, 12| percent. 2. $750. 5. $500. 6. B-A 7. $252.13. $330.92. $416.95. 8. 57|, 32 nearly. 9. $496.97. 10. 48.2 years nearly. 11.- ^°ff2 log {mn - m + n) - log mn 13. $1.00, $23912 (lO)**. 12. 17.67. EXERCISE XXXIV. (Paok 323.) 1. $374.11. 2. $3137.14. 3. 4| per cent. 4. 28f. 5. $3115.55. 6. $3385.20. 7 9. $7360.08, $6404.74, $2901.83. 10. ^. j2 100(1.06^-490 jq.06yo_(io6)j^ ■ (1.06)*- 1 -+ ^" l~{(T.06r7i}F-| '2.42. 8. $2199.95. 13. 10.74. 14. .£1308 12s. 4|d. 1.05 -(1.06)* '{ ■) 15. $4200. 16. 19. 21. log(w-l) log (1+r)" 18. n \n^\) {2 + (w-l)r}. I (^) " - i} 20. ^(^i!L±l.) - (l±!)!_-_-^ 'l\^/ r\ 2 (l+r)"-V^ P-Q log w - log (w - 1) log (n- 1) - log (ri - 2^ log 2 } log(l+r) log(l+r) log(l+r)* MISCELLANEOUS EXAMPLES. EXERCISE XXXV. (Paoi 827.) 1. l+x + x^ + r^ + x* + .r^ + a^ + x'' + ci^ + si!'. ^ a" b' , a h 8. a* = n{J)C - a*), y = n{ca - Jr), z = n{ah - c^). 4. a*- 1. 7. {a + by Hf-zx) ^z^-xy) X + y^ + ?^ - ^xyz ir^ + if -v ^ - Swy. \txijz, 10. -2 r- 1, 2r-l 27(.«;^ - ?/;s) (y/ - zx) {z- - a-y) (.r + f + z^ - Sxyzf scales of f), 8, 11, etc > "■'} '■'■) 19. 22. (.r-^)(.r-<^) {x + a){x + b)(x + cy 20. 999,2220. 21. c? 2d+r I'.M + l. X \_(x_ -y\ (l-.f«) iy(l-2/'r 1-ar y 23. ,2n-a' (jr + yy 18 24. -^(6»-l), arV'-l) Mw+l)(2w+l) {xyY+^-(xyy x'-l nj -1 .(6») - 2" + 25 a • ** • ■=1: v'2:'^\+ V + V2. 6.5 26. 2(a + b) m'.r- + 1 2 ^3? + 4r) ar* + 4/>.'i'^ + 6^.r- + \rx + < 91. wi*** term = Irn^ -vn? — 2m + 1 sum = 2 {n{n-\-\)y w(n+l)(2w+l) 1 94. (1) a' = f/ and aar' + r/.r-l=0; (2) a? = 2, - -(n)(»^+l)+«. 1 97. «i«-l) tti '-1 102. 480. 99. -1+ v^-l, -1- -vZ-l, -3, 1. 103. 1 (n+l)(n + 2)(2w + 3) 108. = 118. - p* + ip\ - 9>pr ^ -ji- nx 380. n J_1__L _i_ L}1 ')■ yo ^^-^^^ '^il^ ^tJi^Z^y^^^iy^ /i^>*»_ >*A >^^ r y actual multiplication, that (./; + a){x + b) = x^-\- (a + h)x + a&, (j* + a){x + h){x + c) = .c^ + (a + /-» + c)x- + {nh + ._,.r"-^ + etc. + ;>„_ i where j^j s= Oj + (/^ + «3 + etc. , ^^2 = ttiMj + ai«3 + «2«3 + otc, etc. = etc., Then, multiplying by another factor, x + <«„, wo have (a: + a,)(j: + (/,).... (.f + ^'„) = j;» +j[)ii;"-' +^>»2.K"-''' + etc. +Pn-it + a„a:"-^ +/>i«„.''""''^ + etc. +i;„_2«„.r +7>„_ifl'„ = .r" + giX"- ' + qrft"- ■ '^ + etc. + (?„ . i« + a^, ^3, etc., a„, taken one, two, three, etc., together; and, consequently, the number of terms in y^ is Ci, in q^ in C.^, etc., where C^, Cj — (7„, represent I IMAGE EVALUATION TEST TARGET (MT-S) 1.0 I.I 1.25 |30 '™^ 2.5 22 2.0 U IIIIII.6 Photographic Sciences Corporation 33 WEST MAIN STREET WEBSTER N.Y. 1456 J (716) 872-4503 iV iV St :\ \ *^ - %\rr each of ch of the ^ely; and >