Question

If $S_n$ denotes the sum of first $n$ terms of a non-constant A.P. whose first term is $a$ and $\frac{S_{nx}}{S_x}$ is independent of $x,$ then $S_p$ is equal to

• A. $p^3$
• B. $p^{2}a$
• C. $p{a}^2$
• D. $a^3$

Answer 1

The correct option is B $p^{2}a$

Let $d$ be the common difference.
$\frac{S_{nx}}{S_x}=\frac{\frac{nx}{2}[2a+(nx-1\left)d\right]}{\frac{x}{2}[2a+(x-1\left)d\right]}=\frac{n\left[\right(2a-d)+nxd]}{(2a-d)+xd}$

For $\frac{S_{nx}}{S_x}$ to be independent of $x$,
$2a-d=0\Rightarrow 2a=d$

Hence, $S_p=\frac{p}{2}[2a+(p-1\left)d\right]$
$=p^{2}a$