Question

Derive the formula for the sum of the first n odd positive integer numbers.

Answer 1

The sequence of the first n odd positive integer numbers is $1,3,5,7,9,...,2n-1.$
$a_1=1,d=2$
We know that $S_n=a_1+a_2+a_3+....a_n=\frac{n}{2}[2a_1+(n-1\left)d\right]$
$S_n=\frac{2n-1}{2}\left[\right(2\times 1+2n-1-1)\times 2]$
$S_n=\frac{2n-1}{2}\left[\right(2n)\times 2]$
$S_n=2n(2n-1)$
$S_n=4n^2-2n$