Mathematics - Sequence & Series Question with Solution | TestHub

Given, $a_1=3,m=5n$ and $a_1,a_2,\dots$ are in AP. $\therefore \frac{S_m}{S_n}=\frac{S_{5n}}{S_n}$ is independent of $n$. Now, $\frac{\frac{5n}{2}[2\times 3+(5n-1)d]}{\frac{n}{2}[2\times 3+(n-1)d]}$ $\Rightarrow \frac{f\{(6-d)+5n\}}{(6-d)+n}$ independent of $n$, if $\begin{array}{rl}6-d=& 0\Rightarrow d=6\\ \therefore a_2=& a_1+d=3+6=9\\ & \text{ Or }\end{array}$ If $d=0$ $\frac{S_m}{S_n}$ is independent of $n$. $$\therefore a_2=3$$