How to prove for each positive integer $n$, the sum of the first $n$ odd positive integers is $n^2$? I'm new to induction so please bear with me. How can I prove using induction that, for each positive integer $n$, the sum of the first $n$ odd positive integers is $n^2$?
I think $9$ can be an example since the sum of the first $9$ positive odd numbers is $1,3,5,7,9,11,13,15,17 = 81 = 9^2$, but where do I go from here. 

We have, (x+1)^2 - x^2 = 2x + 1
Using this, we can say,
1^2 = 0^2 + 1 = 0 +1 = 1
2^2 = 1^2 + 3 = 1 + 3 = 4
3^2 = 2^2 + 5 = 1 + 3 + 5 = 9 and so on...
This explains why the sum is n^2.
