Question

The $n^{th}$ term of an A.P is $p^2$ and the sum of the first $n$ terms is $s^2$. The first term is

• A. $\frac{p^{2}n+2s^2}{n}$
• B. $\frac{p^{2}n+2s^2}{n^2}$
• C. $\frac{p{s}^2-p^{2}s}{n}$
• D. $\frac{2s^2-p^{2}n}{n}$

Answer 1

The correct option is D

$\frac{2s^2-p^{2}n}{n}$

Find the given term for the arithmetic progression

It is given that,

The $n^{th}$ term of an A.P is $p^2$

The sum of the first $n$ terms is $s^2$

We know the sum of the first $n$ terms is $\frac{n}{2}\left(\mathrm{First}\mathrm{term}+\mathrm{Last}\mathrm{term}\right)$

$$\begin{gathered} \therefore s^2=\frac{n}{2}\left(a+p^2\right) \\ \Rightarrow 2s^2=na+n{p}^2 \\ \Rightarrow na=2s^2-n{p}^2 \\ \Rightarrow a=\frac{2s^2-n{p}^2}{n} \end{gathered}$$

Hence, the correct option is $\left(\mathrm{D}\right)$.