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Question

Find sum: 1+5+12+22+35+.....

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Solution

Let
Sn=1+5+12+22+35+.........Sn= 1+5+12+22+......____________________________
tn=1+4+7+10+13+.......n terms
tn=n2{2×1+(n1)3}
=n2(2+3n3)
=n2(3n1)
tn=3n223n22n2
Sn=nn=1tn
=32×n(n+1)(2n+1)612n(n+1)2
=n(n+1)4[3(2n+1)31]
=n(n+1)4×2n
Sn=n2(n+1)2
Hence, Sn=n2(n+1)2

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