The sum of the first n  terms of the series 12+2 . 22+32+2 . 42+52+2 .62+..........  is n(n+1)22,  when n  is even. When n  is odd the sum is

When n is odd, last term will be n2 Then, the sum is  12+2 . 22+32+2 . 42+52+2 .62+..........+2(n−1)2+n2                          =(n−1)n22+n2[Replacing  n  by  n−1  in  n(n+1)22]                           =n3−n2+2n22=n3+n22=n2(n+1)2