_ k=1 ^ 2 n+1 (-1)^ k-1 k^2 equals to

Step-by-step solution

aligned & _ k=1 ^ 2 n+1 (-1)^ n-1 k^2 & = [1^2-2^2+3^2-4^2+ -(2 n)^2+(2 n+1)^2 ] & = [1^2+3^2+5^2+ +(2 n+1)^2 ] & - [2^2+4^2+6^2+ +(2 n)^2 ] & = [1^2+2^2+3^2+4^2+ +(2 n+1)^2 ] & -2 [2^2+4^2+6^2+ +(2 n)^2 ] & = (2 n+1)^2-2 2^2 [1+2^2+3^2+ +n^2 ] & = (2 n+1)(2 n+1+1) 2(2 n+1)+1 6 - n^2 & [ n^2= n(n+1)(2 n+1) 6 ] & = (2 n+1)(2 n+2)(4 n+3) 6 & -8 n(n+1)(2 n+1) 6 & = (n+1)(2 n+1) 6 [2(4 n+3)-8 n] & = (n+1)(2 n+1) 6 [8 n+6-8 n] & =(n+1)(2 n+1) & aligned