Binomial

b c-Fd+8;c.

c . d+&c.

In the same manner, C a .b c .-Fd+&c.

± a . c . d-l- &c.

b . c . &c.

and so on ; that is, A is the sum of the quantities a, b, c, &c. B is the sum of the products of every two; C is the sum of the products of every three, &c. &c.

Let a = b = c = d = &c. then A, or a + b c (1+ &c. = na; = ab ac B + be = a' X the number of combinations of a, b, c, d, &c. taken two n-1 and two, =n .— 2 in the same manner it appears that C = n 271=2.a3 &c. +a . x+b • tit+c . &c. to n factors x a); n = xn+n n —1 It —1 d n . xn—' + 71. ••---•••• • 2 n— + &c.

This proof applies only to those cases in which n is a whole positive number; but the rule extends to those cases in which n is negative or fractional.

Ex 1. = x+ 28 ad x" + 56 at X3 +70 a4 + 56 a3 xs + 28 a' + 8 a Ex. 2. 1 n = -1-7Z • 71— n — 2 +n J Er. 3. = a' +na" n — 2 la' n-4 x4+ &c.

If either term of the binomial be nega. tive, its odd powers will be negative, anc consequently the signs of the terms, it which those odd powers are found, wil be changed.

Ex. 4 = — 8 at x+28 X —56 as x3 +70 al x3-56 a3 x3+ 28 a' x( —8 a -I- Ex. 5. a' n--' x' n — 1 . 2 —a" If the index of the power to which binomial is to be raised be a whole posi tive number, the series will terminate, be n 1 n 3 — 2 cause the co-efficient n .

3 &c. will become nothing when it is con tinned to +I factors. In all othe cases the number of terms will be indefi nite.

When the index is a whole positive number, the coefficients of the terms to ken backward, from the end of the are respectively equal to the co-efficient of the corresponding terms taken for ward from the beginning.

Thus, in the first example, where a+z is raised to the 8th power, the co-efficient are, 1, 8, 28, 56, 70, 56, 28, 8, 1.

In general, the co-efficient of the term is n.m—d.n —2 . 3.2.1 1.2.3 n-2 . n-1 .n The co-efficient of the nth term is 1.2.3..

= n ; of the n—lth tern n-2.n-1 n . n—l.n—i.. .3. n .n &c.

1 . 2 . 3 ..... n— 2 1 . 2 The sum of the coefficients 1+24n.

22 — 1 + &C. is 2rt.

2 For if x=a=1, then n =1+1 n =an= —1 + &c.

Since x-P; n = .4- n a n .

2 1 — a' + &c.

n-1 And x7—T:In=xn—n a 2 a' x++ '—&c.

By addition,7-ra)n 2 . +2 . n 2 + &c.

ts , n-1 xn-t-n .— 2 2 a' ± &c.

Bysubtracting one seriesfrom the other, n — 1 n a + n .

2 2 — &c.

3 The trinomial a-l-b+c may be raised to any power by considering two terms as one factor, and proceeding as before.

Thus, a+b Rn.--.an-Fn . b+c . n1 +n . — . &c. and 2 the powers of b+c may be determined by the binomial theorem.