Question

The sum of $1+3+5+7+..........$ upto $n$ terms is

• A. ${\left(n+1\right)}^2$
• B. ${\left(2n\right)}^2$
• C. $n^2$
• D. ${\left(n-1\right)}^2$

Answer 1

The correct option is C

$n^2$

Explanation for the correct option:

Given, $1+3+5+7+..........$

$1+3+5+7+..........$ is an arithmetic progression with first term $a=1$ and common difference $d=2$

Sum of an arithmetic progression$=\frac{n}{2}\left[2a+(n-1)d\right]$

Sum of arithmetic progression $1+3+5+7+..........$$=\frac{n}{2}\left[2\times 1+(n-1)\times 2\right]$

$$\begin{gathered} =\frac{n}{2}\left(2+2n-2\right) \\ =\frac{n}{2}\times 2n \\ =n^2 \end{gathered}$$

Hence, C is the correct option.