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Question

n2+2n

Solution
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an=n2+2nSn=nk=1(k2+2k)=nk=1k2+nk=12k........(1)
Consider nk=12k=21+22+23+...
The above series 2,22,23,... is a G.P. with both the first term and common ratio equal to 2.
nk=12k=(2)[(2)n1]21=2(2n1)(2)
Therefore, from (1) and (2), we obtain
Sn=nk=1k2+2(2n1)=n(n+1)(2n+1)6+2(2n1)

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