The coefficients of the r^ th , (r+1)^ th and (r+2)^ th terms in the binomial expansion of (1+2x)^n are in the ratio 1 : 4 : 12 . Find the term independent of x in the expansion of (x^2 + 1 x )^ n-r .

Step-by-step solution

The general term in the expansion of (1+2x)^n is T_ k+1 = ^ n C_ k (2x)^k . The coefficients of the r^ th , (r+1)^ th and (r+2)^ th terms are ^ n C_ r-1 2^ r-1 , ^ n C_ r 2^r and ^ n C_ r+1 2^ r+1 respectively. Given the ratio is 1 : 4 : 12 , we have: ^ n C_ r 2^r ^ n C_ r-1 2^ r-1 = 4 1 n-r+1 r 2 = 4 n-r+1 r = 2 n - r + 1 = 2r n - 3r + 1 = 0 (1) And, ^ n C_ r+1 2^ r+1 ^ n C_ r 2^r = 12 4 = 3 n-r r+1 2 = 3 2n - 2r = 3r + 3 2n - 5r - 3 = 0 (2) From equation (1), n = 3r - 1 . Substituting this into equation (2): 2(3r