Question

Let $a_1,a_2,a_3,\dots \dots ,a_{100}$ 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 equal to___

Answer 1

Given, $a_1=3,m=5nand{a}_1,a_2,\dots \dots$ is an AP
$\therefore \frac{S_m}{S_n}=\frac{S_{5n}}{S_n}$ is independent of n
$=\frac{\frac{5n}{2}[2\times 3+(5n-1\left)d\right]}{\frac{n}{2}[2\times 3+(n-1\left)d\right]}=\frac{5\left\{\right(6-d)+5nd\}}{(6-d)+nd}$, independent of n
If $6-d=0$
$\Rightarrow d=6$
$\therefore a_2=a_1+d=3+6=9$
or If d = 0, then $\frac{S_m}{S_n}$ is independent of n.
$\therefore a_2=9$