SATHEE: Sequences and Series 2 Question 22

Sequences and Series 2 Question 22

23. Let $a_{1}, a_{2}, a_{3}, \dots, a_{100}$ 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 $a_{1}, a_{2}, \dots$ , is an AP.

$∴ \frac{S_{m}}{S_{n}} = \frac{S_{5 n}}{S_{n}}$ 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 $6 - d = 0 \Rightarrow d = 6$

independent of $n$

$∴ a_{2} = a_{1} + d = 3 + 6 = 9$

or If $d = 0$ , then $\frac{S_{m}}{S_{n}}$ is independent of $n$ .

$∴ a_{2} = 9$

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Sequences and Series 2 Question 22

23. Let a1,a2,a3,…,a100 be an arithmetic progression with a1=3 and Sp=∑i=1pai,1≤p≤100. For any integer n with 1≤n≤20, let m=5n. If SmSn does not depend on n, then a2 is equal to ……

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23. Let $a_{1}, a_{2}, a_{3}, \dots, a_{100}$ 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 ……