Typically in a mathematical induction problem you prove the statement for some specific value of k, then use the theorem itself to simplify your problem.

Specifically in your case we know that the theorem is true for k = 1.

Let me change your variables a bit. (This may make this more transparent, or it may confuse you. I hope the former!) Assume the theorem is true for some k = n. Then

We need to show that the theorem is true for the next value of k, n + 1. That is, we need to show that:

Break down the sum on the LHS:

The sum in parenthesis is just , according to our assumption. So we have:

which is an identity, so the theorem is true for k = 1, thus k = 2, 3, 4, ...

-Dan