Let a 1 , a 2 , a 3 , … , a 100 be an arithmetic progression with a 1 = 3 and S p = i = 1 p a i , 1 p 100 . For any integer n with 1 n 20 , let m = 5 n . If S m S n does not depend on n , then a 2 is [2011]

Question

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 [2011]

Smart Achievers · Accepted Answer

(9)

$\frac{S_{m}}{S_{n}} = \frac{S_{5 n}}{S_{n}} = \frac{\frac{5 n}{2} [2 \times 3 + (5 n - 1) d]}{\frac{n}{2} [6 + (n - 1) d]}$ $[∵ m = 5 n]$

$= \frac{5 [(6 - d) + 5 n d]}{(6 - d) + n d},$ which will be independent of $n$ if $d = 6$ or $d = 0$

For a proper A.P., we take $d = 6$

$∴$ $a_{2} = a_{1} + d = 3 + 6 = 9$

Solution

Q.
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 [2011]

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Solution

Q. 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 [2011]

Ans. (9) SmSn=S5nSn=5n2[2×3+(5n-1)d]n2[6+(n-1)d] [∵m=5n] =5[(6-d)+5nd](6-d)+nd, which will be independent of n if d=6 or d=0 For a proper A.P., we take d=6 ∴ a2=a1+d=3+6=9

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Q.
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 [2011]