Question

(2) If the coefficients of (2r + 4)th, (r - 2)th terms in the expansion of $(1 + x)^18$ are equal, find r.

Answer 1

Given expansion is $(1 + x)^(18)$
Now, $(2r + 4)$ th term i.e., $T(2r + 3 + 1)$
$:. T(2r+ 3 + 1) = .^(18)C(2r + 3) (1)^(18 - 2r - 3) (x)^(2r + 3)$
$= .^(18)C(2r + 3) x^(2r + 3)$
Now, $(r - 2)$ th term i.e., $T(r - 3+ 1)$
$:. T(r - 3 + 1) = .^(18)C(r - 3) x^(2r + 3)$
Now, $(r - 2)th$ term i.e., $T(r - 3 + 1)$,
$:. T(r - 3 + 1) = .^(18)C(r - 3) x^(r - 3)$
As, $.^(18)C(2r + 3) = .^(18)C(r - 3) [:. .^(n)C_(x) = .^(n)C_(y) rArr x + y = n]$
$rArr 2r + 3 + r - 3 = 18$
$rarr 3r = 18$
$:. r = 6$