014 



M ATHEMATICa ALGEBRA. 



[THE BINOMIAL THEOREM. 



Thus the coefficient of the term which in- II. n 

 volvo* a* - 'tr is 



1- -2. 3 ..... r " 



9. To prow A Binomial TVorwm wfon n u 



(a) Toshow that (a + 6)- 

 By actual multiplication, 

 + &) - o 



a + 6) - a + 4a s & + ____ 

 These results plainly suggest the assumption 



(o + 6)- - a" + ma--6 + . . . . 

 Multiply both sides by a + 2, and we have 



(a + &r * > - a"* 1 + (m + 1) a-6 + . . . . 

 which is clearly of the same form as the assumption, i. e., 

 this has m + 1, wherever that has m. Hence, if the 

 theorem be true for m, it must also be true for m + 1. 

 Now it is true for 4, .'. it is true for 5, .' . for 6, and 

 so on ; then-fore it Ls always true for any positive 

 whole number. 



..(o + &)" = o" + tu J 6 + ---- (a) 

 N.B. If o = l, and & *> x, we of course have 



' 



(ft) To show that 





For suppose 

 (l+z)-=l + nz + A 2 *' + A 8 x + A 4 * + ... (c) 



It is plain, since (1 + z)" means (1 + x) multiplied 

 into itself n times, that this series is finite, so that we 

 may employ the principle of indeterminate coefficients. 



In the series A 2 , A 3 , A 4 , ..... do not at all depend 

 on z, and will therefore continue the same for all values 

 of x, so that, for instance 



(I + y)" = 1 + *y + A 2 y +A 5 y + A 4 y* + . . . . 



In equation (o), write x = y + z. Then 

 (1 + *+*)" = 1 + (y + *) + A, (y + *)' + A, (y + ,) 



= 1 + .y + + A, (y' + 2yz + .. . ) + A, (y> + 3y-2 



+ &c. (<f) 



a and z = b, 



+ *{. + 2A,y + 3A 3 y + 4A 4 y' +... 

 which is true for all values of y and z. 



Again, if in equation (a) we take 1 + y 

 we nave 



OTy + *)=(!+*) +n(l + y) -z+. . . . (E). 

 Now, (d) and (e) are the same for all values of 2, 

 .*. the coefficient of z in each must be the same, .'. 



(1 + y). - ' = + 2A 3 y + 3A 3 y' + 4 A 4 y" + . . . . 

 for all values of y ; multiply both sides by 1 + y, 



But by equation, (c) n (I + y)= n + n'y + A a ny 5 + A 3 ny 3 

 -f A 4 ny< + 



. . -}-ii*y 4-A,y* + nA 3 y > -j-nA 4 y 4 + .... 

 = + (2 A, +)y-r-(3A, -\- 2Aj)y* + (4A 4 -|- 3A 3 )y'-|-. . . . 

 These expressions are true for all values of y. 

 Hence 2A, -f n = n 1 



3A 3 -f 2A a - nA, 

 4A 4 -f- 3A 8 - nA 8 



.'. 2A, - n n - n(n 1(, .-. A, - ^j^ * 



SA, - A.(n - 2) A, -?&=M^> 

 4A 4 -A,(n 3) A,- 



(i). 



The student will observe the manner in which each 

 successive coefficient is derived from the one that goes 

 before it He will easily see that if we look in the r 

 1" and I-" 1 terms viz., A r _,. x,_! + A^v + . . . . 

 we should then have an equation 



r\, (r 1) A P _,-nA r _,, 

 .'. rA r = A,_,(n r + 1). 



so that the general term of the expansion will be 

 n.(n l)(n 2). . . (n r + 1) 

 1' 2- 3- ... r 



He will also observe that if n be a whole number 

 when r is greater than n, there will be in the general 

 term a factor n n -f- 1 -f- 1 = o ; or x" ia the last 

 term of the series, also the coef. of 

 ^ n.n 1 ... 2-1 



x> + ...+* 

 (7) To prove the series 



(0 



. 



For in the last series write x =, _. Then- 



a 



But 



X a" 



. . , 



which b the Binomial Theorem when n is a positive 

 integer. 



In page 461, a table of the developments of powers of 

 a binomial is given. These may be immediately deduced 

 from the series we have just proved. Thus, to develop, 

 or expand (a + z) 8 , we have 



... (a 



= a" + 8 a'x 





1-23-4 



a4 

 ' 



,87-65-4 

 * 1-2 3 -4-r, 

 . 87-6-0-4-3-2 



7'6-5-4-3. 



+ ^L 



1-2 -3 -4 -6 -6 



, 87-6 -6 -4 -3 -2-1 . 



1-2-3 -4 -6 -67-8' 

 , , 87-6 .. 87-6-6 

 h 1-2-3 ~ 1 2.3.4' 



-o' + 8a'sc + 28oV -f 



+ 28ax + Sax' + x'. 



Tlie student will observe that the coefficients of o 7 x 

 and of ox 1 are the same, as also of o"x* and a 2 *" of o 5 * 3 

 and aV. 

 And, in general, if wo write the series, whether we 



