In the binomial expansion of (1 + ax)^n , let A_r be the coefficient of x^r . If _ r=1 ^ n r ( A_r A_ r-1 ) = 72 and _ r=1 ^ n r^2 ( A_r A_ r-1 )^2 = 816 , then the coefficient of x^3 in the expansion is equal to

Step-by-step solution

The general term in the expansion of (1 + ax)^n gives A_r = ^ n C_r a^r . The ratio of consecutive coefficients is: A_r A_ r-1 = ^ n C_r a^r ^ n C_ r-1 a^ r-1 = a ( n-r+1 r ) For the first sum: _ r=1 ^ n r ( a n-r+1 r ) = a _ r=1 ^ n (n-r+1) a (n + (n-1) + + 1) = a n(n+1) 2 = 72 For the second sum: _ r=1 ^ n r^2 ( a n-r+1 r )^2 = a^2 _ r=1 ^ n (n-r+1)^2 a^2 (n^2 + (n-1)^2 + + 1^2) = a^2 n(n+1)(2n+1) 6 = 816 Squaring the first equation gives: a^2 n^2(n+1)^2 4 = 5184 Dividing this by the second equation to eliminate