Find a formula for $1 + 3 + 5 + ... +(2n - 1)$, for $n \ge 1$, and prove that your formula is correct. I think the formula is $n^2$.
Define $p(n): 1 + 3 + 5 + \ldots +(2n − 1) = n^2$
Then $p(n + 1): 1 + 3 + 5 + \ldots +(2n − 1) + 2n = (n + 1)^2$
So  $p(n + 1): n^2 + 2n = (n + 1)^2$
The equality above is incorrect, so either my formula is wrong or my proof of the implication is wrong or both.
Can you elaborate?
Thanks.

$p(n): 1 + 3 + 5 + \ldots +(2n − 1) = n^2$
$p(n + 1): 1 + 3 + 5 + \ldots +(2n − 1) + 2n+1 =n^2+2n+1= (n + 1)^2$
next term after $2n-1$ is $2n+1$ is not $2n$ as you mean
