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Question

If total number of runs scored in n matches is (n+14)(2n+1n2) where n>1, and the runs scored in the kth match are given by k2n+1k where 1kn, then the value of n is

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Solution

Let Sn=nk=1k2n+1k=2n+1nk=1k2k

Let tn=nk=1k2k
tn=12+222+323++n2n (1)
tn2= 122+223++n2n+1 (2)

From (1)(2), we get
tn2=12+122+123++12nn2n+1
tn2=12(1(12)n)112n2n+1
tn=2(112nn2n+1)

Sn=2n+1×tn
=2n+1×2(112nn2n+1)
=2[2n+12n]
On comparing with the expression (n+14)(2n+1n2),
n+14=2n=7

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