Sequences and Series 2 Question 22
23. Let $a _1, a _2, a _3, \ldots, a _{100}$ be an arithmetic progression with $a _1=3$ and $S _p=\sum _{i=1}^{p} a _i, 1 \leq p \leq 100$. For any integer $n$ with $1 \leq n \leq 20$, let $m=5 n$. If $\frac{S _m}{S _n}$ does not depend on $n$, then $a _2$ is equal to ……
(2011)
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Solution:
- Given, $a _1=3, m=5 n$ and $a _1, a _2, \ldots$, is an AP.
$\therefore \quad \frac{S _m}{S _n}=\frac{S _{5 n}}{S _n}$ is independent of $n$.
$=\frac{\frac{5 n}{2}[2 \times 3+(5 n-1) d]}{\frac{n}{2}[2 \times 3+(n-1) d]}=\frac{5{(6-d)+5 n}}{(6-d)+n}$,
If $\quad 6-d=0 \Rightarrow d=6$
independent of $n$
$\therefore \quad a _2=a _1+d=3+6=9$
or If $d=0$, then $\frac{S _m}{S _n}$ is independent of $n$.
$\therefore a _2=9$
