Question

Let $a_{1}, a_{2}, a_{3}, \ldots, a_{100}$ be an arithmetic progression with $a_{1}=3$ and $S_{p}=\displaystyle\sum_{i=1}^{p} a_{i}, 1 \leq p \leq 100 .$ For any integer $n$ with $1 \leq n \leq 20$, let $m =5 n$. If $\frac{ S _{ m }}{ S _{ n }}$ does not depend on $n$, then $a _{2}$ is_______

Answer 1

$a _{ l }, a _{2}, a _{5}, \ldots a _{100}$ is an A.P.
$a_{1}=3, S_{p}=\displaystyle \sum_{i=1}^{p} a_{i}, 1 \leq p \leq 100$
$\frac{S_{m}}{S_{n}}=\frac{S_{5 n}}{S_{n}}=\frac{\frac{5 n}{2}(6+(5 n-1) d)}{\frac{n}{2}(6-d+n d)}$
$\frac{S_{m}}{S_{n}}$ is independent of n of $6-d=0 \Rightarrow d=6$