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Question

The sum of n terms of the series 312+512+22+712+22+32+_________ is _______________.

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Solution

Sum of n term of 312+512+22+712+22+32
here nth term tn for above series is
i.e Tn=2n+112+22 + .... +n2Tn=2n+1nn+12n+16=2n+1n+1 2n + 16Tn = 6nn+1i.e Tr= 6rr+1
Hence r=1n Tr= r=1n 6rr+1 = 6 r=1n 1rr+1=r=1n6 1r - 1r+1= 6 11 - 12 + 12 - 13 + 13 - 14 +....+ 1n - 1n+1 = 6 1 - 1n + 1 =6 nn+1
Hence, Sn sum of n term = 6nn+1

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