Perhaps this will help, perhaps it will not.

The first equation (N = 1) is given as

The second (N = 2) is

The third (N = 3) is

etc.

Notice that the lowest number for n is given by the series 1, 2, 5, 10, 17, etc. This is the recursive function

where N is the equation number. So we have:

The first equation (N = 1) is given as

The second (N = 2) is

The third (N = 3) is

etc.

Now notice that the upper term of the summation is of the form f(N) + g(N) where g(n) = 2(N - 1). So the general Nth equation is given by

This is probably a more complicated method than is needed, but it's the one I came up with that was the most demonstrative. Anyway, now solve f(N) as a function of N (rather than by recursion) and I'd guess the rest would be a simple induction proof.

-Dan