Question

If the $n^{th}$ term of an A.P be $(2n-1)$, then the sum of its first $n$ terms will be

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

Answer 1

The correct option is C

$n^2$

The explanation for the correct answer.

Step 1: Find the first term and common difference.

Let $a$ the first term and $d$ be the common difference of the A.P

Given that: The nth term of the AP is $\left(2n-1\right)$

The formula for the nth term of AP is $a_n=a+(n-1)d$

$$\begin{gathered} a_n=a+(n-1)d=(2n-1) \\ {a}_1=(2\times 1-1)=1 \\ {a}_2=\left(2\times 2-1\right)=3 \\ d=a_2-a_1=3-1=2 \end{gathered}$$

Step 2: Find the sum of the terms of AP

Sum of the terms of AP is $S_n=\frac{n}{2}\left[2a+(n-1)d\right]$

$$\begin{gathered} S_n=\frac{n}{2}\left[2\times 1+(n-1)\times 2\right] \\ {S}_n=\frac{n}{2}\left[2+2n-2\right] \\ {S}_n=\frac{n}{2}\left[2n\right]=n^2 \end{gathered}$$

Hence, option (c) is correct.