Let a_1, a_2, a_3, ... ... , a 100 be an arithmetic progression with a_1=3 and S_p=Sigma_i=1^p a_i, 1 <= p <= 100. For any integer n with 1 <= n <= 20, <= t m=5 n. If (S_m / S_n) does not depend on .n then a_2 is \_\_\_\_ .

Let a_{1},a_{2},a_{3},.,a_{n} be an arithmetic progres... | Sequence & Series (Mathematics)

$\frac{s_m}{s_n}=\frac{s_{5 n}}{s_n}=\frac{\frac{5 n}{2}(6+(5 n-1) d)}{\frac{n}{2}(6+(n-1) d)} \\ = \frac{5((6-d)+5 n d)}{(6-d)+n d}$ $\\$ For $d=6$ we have $S_{5 n}/S_n$ independent of ' $n$ '. $\\$ Also, For $d=0$ , it's trivially true. So, $a_2=9$ $\\$ Note : This question has two possible answers. $\\$ So although the question is correct, answer is not unique

Explanation

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