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, ...