Question

Let $a_1,a_2,a_3,\dots ,a_{100}$ be an arithmetic progression with $a_1=3$ and $S_p$ is sum of 100 terms . For any integer $n$ with $1\le n\le 20$, let $m=5n$. If $\frac{S_m}{S_n}$ does not depend on $n$, then $a_2$ is

• A. $6$
• B. $7$
• C. $8$
• D. $9$

Answer 1

Answer: (D)

$9$

We know that sum of n terms of an A.P. is given by
$S_k=\frac{k}{2}\{2a+(k-1\left)d\right\}$
$\frac{Sm}{Sn}=\frac{\frac{5n}{2}\{6+(5n-1\left)d\right\}}{\frac{n}{2}\{6+(n-1\left)d\right\}}=5.\left\{\frac{(6-d+nd)+4nd}{(6-d+nd)}\right\}=5.\{1+\frac{4nd}{6-d+nd}\}=5.\{1+\frac{4d}{\frac{(6-d)}{n}+d}\}$
Since this is independent of $n$, then $6-d=0=>d=6$
Hence, the second term = first term $+6=3+6=9$
Hence, D is correct.