# Indirect proof that sum of first n even numbers is $n^2 + n$

I'm learning about proof by contrapositive and by mathematical induction in a computer science class. I'm banging my head trying to solve this problem and would like some help:

Prove that the sum of the first $n$ even numbers is $n^2 + n$
(a) indirectly by assuming that the sum of the first $n$ odd numbers is $n^2$
(b) directly by mathematical induction.

I have no problem doing (b). But I can't figure out how to do (a) using indirect proof. I can only come up with this:

Sum of first $n$ numbers is $n^2 + n \implies$ numbers are even

Contrapositive: numbers are odd $\implies$ sum of first n numbers is not $n^2 + n$.

This just doesn't seem logical to me. Any hints would be greatly appreciated.

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The picture here may give you some ideas. – Larry Wang Sep 21 '10 at 0:46

Using the contrapositive isn't the only indirect way of proving a theorem, and I highly doubt it's what your instructors had in mind here. As a hint for part (a): what can you say about the first $n$ even numbers (individually) compared to the first $n$ odd numbers? Can you find some convenient correspondence between, say, the 5th even number and the 5th odd number?