If the coefficient of x^p in the expansion of (x^2 + 1 x )^n is equal to the coefficient of x^ -p in the expansion of (x - 1 x^2 )^n , where n and p are positive integers such that the respective terms exist, then which of the following must be true?

Step-by-step solution

Let the general term in the expansion of (x^2 + 1 x )^n be T_ r+1 . T_ r+1 = ^ n C_r (x^2)^ n-r ( 1 x )^r = ^ n C_r x^ 2n-3r For the coefficient of x^p , we set 2n - 3r = p r = 2n-p 3 . Coefficient of x^p = ^ n C_ 2n-p 3 . Let the general term in the expansion of (x - 1 x^2 )^n be T_ k+1 . T_ k+1 = ^ n C_k (x)^ n-k (- 1 x^2 )^k = ^ n C_k (-1)^k x^ n-3k For the coefficient of x^ -p , we set n - 3k = -p k = n+p 3 . Coefficient of x^ -p = ^ n C_ n+p 3 (-1)^ n+p 3 . Given that these coefficients are equal: ^ n C_ 2n-p