Let K be the sum of the coefficients of the even powers of x in the expansion of (1+x)^ 2N . Let a be the middle term in the expansion of (2x + 1 x )^ 2N . If ^ 2N C_ N-1 K a = 25 13 2²³ , then the value of N is equal to:

Step-by-step solution

The sum of the coefficients of the even powers of x in the expansion of (1+x)^ 2N is given by K = 2^ 2N-1 . The middle term in the expansion of (2x + 1 x )^ 2N is the (N+1)^ th term, given by: a = , ^ 2N C_ N (2x)^N ( 1 x )^N = , ^ 2N C_ N 2^N Substitute K and a into the given expression: ^ 2N C_ N-1 K a = ^ 2N C_ N-1 2^ 2N-1 ^ 2N C_ N 2^N Using the property ^ n C_ r-1 ^ n C_ r = r n-r+1 , we have ^ 2N C_ N-1 ^ 2N C_ N = N 2N - (N-1) = N N+1 . Thus, the ratio simplifies to: N N+1 2^ N-1 We are given that this ratio