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Let a(1), a(2), a(3),..., a(100) be an a...

Let `a_(1), a_(2), a_(3),..., a_(100)` be an arithmetic progression with `a_(1) = 3 and S_(p) = sum_(i=1)^(p) a_(i), 1 le p le 100`. For any integer n with `1 le n le 20`, let `m = 5n`. If `(S_(m))/(S_(n))` does not depend on n, then `a_(2)` is equal to ....

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The correct Answer is:
9
Given, `a_(1) = 3, m = 5n and a_(10, a_(2),`...., is an AP.
`:. (S_(m))/(S_(n)) = (S_(5n))/(S_(n))` is independent of n
`= ((5n)/(2) [2 xx 3 + (5n -1)d])/((n)/(2) [2 xx 3 + (n -1) d]) = (5 {(6 -d) + 5n})/((6-d) + n)` independent of n
If `6 - d = 0 rArr d = 6`
`:. a_(2) = a_(1) + d = 3 + 6 = 9`
or If d = 0, then `S_(m))/(S_(n))` is independent of n.
`:. a_(2) = 9`
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