Question

Let $a_1,a_2,a_3,\dots ,a_{100}$ be an arithmetic progression with $a_1=3$ and $S_p=\sum _{i=1}^p{a}_i,1\le p\le 100.$ 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_______

Answer 1

Answer: 9

$a_l,a_2,a_5,\dots a_{100}$ is an A.P.
$a_1=3,S_p=\sum _{i=1}^p{a}_i,1\le p\le 100$
$\frac{S_m}{S_n}=\frac{S_{5n}}{S_n}=\frac{\frac{5n}{2}(6+(5n-1)d)}{\frac{n}{2}(6-d+nd)}$
$\frac{S_m}{S_n}$ is independent of n of $6-d=0\Rightarrow d=6$