Let $P(n)$ be : the sum of the first odd natural numbers equal $n^2$. Express in summation notation and use induction.

Question

I presented the first odd $n$ natural integers as $2n+1=n^2$. In summation notation I just factored this into $(n-1)(n-1)$. I am unsure whether this is correct and also I don't know how to carry out induction for this.

Answer 1

A start:

$$P(n): \sum\limits_{i=1}^n (2 i - 1) = n^2$$