Question

# Let a1,a2,a3,……,a100 be an arithmetic progression with a1=3 and Sp=∑pi=1ai,1≤p≤100. For any integer n with 1≤n≤20, 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+(5n−1)d]n2[2×3+(n−1)d]=5{(6−d)+5nd}(6−d)+nd, independent of n If 6−d=0 ⇒d=6 ∴a2=a1+d=3+6=9 or If d = 0, then SmSn is independent of n. ∴a2=9Mathematics

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