IIT-JEE 2011 · Paper 1 · Mathematics Q64

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MathematicsInteger+4Question 64 of 69

Let $a_1,a_2,a_3,\ldots,a_{100}$ be an arithmetic progression with $a_1=3$ and $S_p=\displaystyle\sum_{i=1}^pa_i$ , $1\le p\le100$ . For any integer $n$ with $1\le n\le20$ , let $m=5n$ . If $\dfrac{S_m}{S_n}$ does not depend on $n$ , then $a_2$ is

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Answer and worked solution

Official answer: 9

Ratio. $S_p=\frac p2\left[6+(p-1)d\right]$ , so
$\frac{S_{5n}}{S_n}=\frac{5\left[(6-d)+5nd\right]}{(6-d)+nd}.$

Independent of n when $6-d=0$ (ratio 25) or $d=0$ (ratio 5). So $a_2=9$ or $3$ ; the key accepts both.

Answer: 9 (key: 3, 9, or both)

Topics:Arithmetic progression

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IIT-JEE 2011 · Paper 1, Mathematics question 64