In a binomial expansion of (x+y)^ 2 n+1 (x-y)^ 2 n+1 , the sum of middle terms is zero. What is the value of ( x^2 y^2 ) ?

Step-by-step solution

(x+y)^ 2 n+1 (x-y)^ 2 n+1 Middle term, aligned & 2 n+2 2 , 2 n+4 2 =(n+1) th term, (n+2) th term & T _ n+1 = ^ 2 n+1 C _n (x^2 )^ n+1 (y^2 )^n & ~T _ n+2 =2^ n+1 C _ n+1 (x^2 )^n (y^2 )^ n+1 & ^ 2 n+1 C _n (x^2 )^ n+1 (y^2 )^n=2^ n+1 C _ n+1 (x^2 )^n (y^2 )^ n+1 & 2^ n+1 C _n ^ 2 n+1 C _ n+1 = x^ 2 n y^ 2 n+2 x^ 2 n+2 y^ 2 n & (2 n+1)! (n+1)!n! n!(n+1)! (2 n+1)! = y^2 x^2 & y^2 x^2 =1 & 1 y^2 = 1 1 & x^2 y^2 =1 aligned