The sum of all possible values of n N , so that the coefficients of x, x² and x³ in the expansion of (1+x² )²(1+x)^ n , are in arithmetic progression is :

Step-by-step solution

(1+x^2)^2 = 1 + 2x^2 + x^4 . Multiply with (1+x)^n : Coefficient of x : n . Coefficient of x^2 : n 2 + 2 = n(n-1) 2 + 2 . Coefficient of x^3 : n 3 + 2n = n(n-1)(n-2) 6 + 2n . For AP: 2 ( n(n-1) 2 +2 ) = n + n(n-1)(n-2) 6 + 2n . Simplifying: n^3 - 9n^2 + 26n - 24 = 0 . (n-2)(n-3)(n-4) = 0 n = 2, 3, 4 . Sum = 2 + 3 + 4 = 9 .