Answer
Hence, proved.
$S_n=n^2$.
Work Step by Step
The given sequence is
$1+3+5+....+(2n-1)$.
This is the arithmetic series.
Common difference $d=2$.
First term $a=1$.
Last term $l=(2n-1)$.
Number of terms $n$.
Sum of all terms is $\frac{n}{2}\cdot (a+l)$.
Substitute all values.
$=\frac{n}{2}\cdot (1+(2n-1))$
Simplify.
$=\frac{n}{2}\cdot (1+2n-1)$
$=\frac{n}{2}\cdot (2n)$
$=n^2$.
