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.

Rewrite the series as 
$$\sum\limits_{i=1}^n 2i-1$$
Continuing,
$$...=2\sum\limits_{i=1}^n i - \sum\limits_{i=1}^n 1$$
$$=2\frac{n(n+1)}{2} - n$$
$$=n^2+n-n$$
$$=n^2$$
As you expected.
