# Proof of geometric sum relation by mathematical induction

I understand the concept behind mathematical induction and have worked out some examples before. However, this was given as a question on a homework assignment and I'm unable to work it out. I'm not sure if I'm doing something wrong, but it becomes a bunch of algebra and I'm unable to get the LHS to equal the RHS. Here is the original equation to prove:

$\sum_1^n(-1)^{i+1}i^2 = ((-1)^{n+1}n(n+1))/2$

Correct me if I set this up wrong, but in the inductive step, this is what I am trying to prove:

$((-1)^{n+2}(n+1)(n+2))/2 = (-1)^{n+2}(n+1)^2+(((-1)^{n+1}n(n+1))/2)$

Please let me know if this has been set up correctly, and if so, how to tackle it. No matter what kind of manipulation I try to use, I can't get the two sides to simplify to the same thing.

## 1 Answer

Yes, it is correct.

Now, on the right-hand side, replace $\frac{(-1)^{n+1}(n)(n+1)}{2}$ by $-\frac{(-1)^{n+2}(n)(n+1)}{2}$.

• Thank you so much. That made things a lot easier. My brain just doesn't see things like that :) – Bobazonski Feb 3 '15 at 6:27
• You are welcome. So I take it the bringing to a common denominator and finishing went smoothly. The next time a trick of this nature is needed, you will be ready. – André Nicolas Feb 3 '15 at 6:29