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Question

Let a1,a2,a3,,a100 be an arithmetic progression with a1=3 and Sp=pi=1ai,1p100. For any integer n with 1n20, let m = 5n. If SmSn does not depend on n, then a2 is equal to___


Solution

Given, a1=3,m=5n and a1,a2, is an AP
SmSn=S5nSn is independent of n
=5n2[2×3+(5n1)d]n2[2×3+(n1)d]=5{(6d)+5nd}(6d)+nd, independent of n
If 6d=0
d=6
a2=a1+d=3+6=9
or If d = 0, then SmSn is independent of n.
a2=9

Mathematics

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