<s>[INST] 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. 
 [/INST] Step 1.
For n = 1 it's true that 1 = 2*1-1
Step 2.
We suppose that 1+3+5+...+(2n-1) = n2
and want to prove that: 1+3+5+...+(2(n+1)-1) = (n+1)2
We add (2(n+1) -1) to this:
1+3+5+...+(2n-1) = n2
and get:
1+3+5+...+(2n-1) + (2(n+1) -1) = n2 + (2(n+1) -1)
so:
1+3+5+...+(2(n+1) -1) = n2 + 2n+2 -1
but n2+2n+1 = (n+1)2
so we finally have:
1+3+5+...+(2(n+1)-1) = (n+1)2
</s>