Summing $1,3,5,\ldots$
Perhaps this will help:


$1 = 1$
$1 + 3 = 4$
$1 + 3 + 5 = 9$
$1 + 3 + 5 + 7 = 16$
$1 + 3 + 5 + 7 + 9 = 25$


Can you recognize the sequence $1,4,9,16,25$?

To prove the corresponding formula, use the following idea:
$$
1 + 3 + 5 + 7 = \frac{1+7}{2} + \frac{3+5}{2} + \frac{5+3}{2} + \frac{7+1}{2}.
$$