Page:Elementary algebra (1896).djvu/359

BINOMIAL THEOREM. 341 Ex. 1. Find the expansion of (a + y)^6.

By the formula, the expansion

= a^6 + {}^6C_1 a^5 y + {}^6C_2 a^4 y^2 + {}^6C_3 a^3 y^3 + {}^6C_4 a^2 y^4 + {}^6C_5 a y^5 + {}^6C_6 y^6 = a^6 + 6a^5 y + 15a^4 y^2 + 20a^3 y^3 + 15a^2 y^4 + 6a y^5 + y^6

on calculating the values of 6C1, 6C2, 6C3... .

Ex. 2. Find the expansion of (a — 2x)^7.

$$(a-2x)^7 = a^7 - {}^7C_1 a^6 (2x) + {}^7C_2 a^5 (2x)^2 -{}^7C_3 a^4 (2x)^3 + $$ to 8 terms.

Now remembering that nCr = nCn-r, after calculating the coefficients up to 7C3, the rest may be written down at once; for 7C4=7C3; 7C5 =7C2; and so on. Hence

(a-2x)^7 = a^7 - 7 a^6 (2x) + 7 6 1 2 a^5 (2x)^2 - 7 6 5 1 2 3 a^4 (2x)^3 +... = a^7 - 7a^6 (2x) + 21a^5 (2x)^2 - 35a^4 (2x)^3 + 35a^3 (2x)^4 - 21a^2 (2x)^5 + 7a (2x)^6 - (2x)^7 = a^7 - 14a^6 (2x) + 84a^5 (2x)^2 - 280a^4 (2x)^3 + 560a^3 (2x)^4 - 672a^2 (2x)^5 + 448a (2x)^6 - 128(2x)^7

410. The (r+1)th or General Term. In the expansion of (a + b)^n, the coefficient of the second term is nC1; of the third term is nC2; of the fourth term is nC3; and so on; the suffix in each term being one less than the number of the term to which it applies; hence nCr is the coefficient of the (r+1)th term. This is called the general term, because by giving to r different numerical values any of the coefficients may be found from nCr; and by giving to a and b their appropriate indices any assigned term may be obtained. Thus the (r+1)th term may be written

{}^nC_r a^{n-r}b^r, or n(n-1)(n-2)...(n-r+1)a^{n-r}b^r r

In applying this formula to any particular case, it should be observed that the index of b is the same as the suffix of C, and that the sum of the indices of a and b is n.