Question

Find each sum.

$1+3+5+.........+\left(2n-1\right)$

Short answer

The solution is$S_n=n^2$.

Step-by-step solution

Step 1. Given information

$1+3+5+.........+\left(2n-1\right)$

Step 2. Calculation

We have, $1+3+5+.........+\left(2n-1\right)$

So,localid="1649220172843" $a_1=1,d=2,a_n=\left(2n-1\right)$

Now use formula -

localid="1649220724840" $S_n=\frac{n}{2}\left(a_1+a_n\right)$

localid="1649220732601" $=\frac{n}{2}\left[1+\left(2n-1\right)\right]$

$=\frac{n}{2}\left(2n\right)$

$=n^2$

Therefore, the sum islocalid="1649220855825" $S_n=n^2$.