Discussion Forum

1)

 Let $a_{1},a_{2},a_{3}.......,a_{100}$ be an arithmetic progression with  $a_{1}=3$  and 

$S_{p}=\sum_{i=1}^pa_{i}, 1\leq p\leq100$ for any integer n with $1\leq n\leq 20$ , let m=5n, If $\frac{S_{m}}{S_{n}}$ does not depend  on n, then $a_{2}$ is 

Answer:

Option A,B

Explanation:

Given $a_{1}=3, m=5n$

 and $a_{1},a_{2},....$ 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 

   $6-d=0 \Rightarrow d=6$

 $\therefore$      $a_{2}=a_{1}+d=3+6=9$

 if d=0

 $\frac{S_{m}}{S_{n}}$ is independent of n

 $\therefore$  $a_{2}=3$