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.