Question

Let$a_1,a_2,.............a_{100}$ be an A.P. with $a_1=3$ and $s_p=\sum _{i=1}^{p}ai,a\le p\le 100$ for any integer n with $1\le n\le 20$, let $m=\sqrt{n}$. If $\frac{s_m}{s_n}$ does not depend on n then $a_2$ is

• A. 9
• B. 0
• C. 1
• D. 18

Answer 1

Answer: (A)

9

$\frac{s_m}{s_n}$=$\frac{s_{5n}}{s_n}$=$\frac{\frac{5n}{2}[6+(5n-1\left)d\right]}{\frac{n}{2}[6+(n-1\left)d\right]}$=$\frac{5\left[\right(6-d)+5nd]}{\left[\right(6-d)+nd]}$
d=6 or d=0
If d=0 then $a_2$ =3 else $a_2$ = 9
For single choice, more appropriate choice is 9
$\therefore a_2=3+6=9$