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Question

12+(12+22)+(12+22+32)+....

Solution
Verified by Toppr

The given series is,12+(12+22)+(12+22+32)+...an=(12+22+32+...+n2)=n(n+1)(2n+1)6=n(2n2+3n+1)6=23+3n2+n6=13n3+12n2+16nSn=nk=1ak=nk=1(13k2+12k2+16k)=13nk=1k3+12nk=1k2+136nk=1k=13n2(n+1)2(2)2+12×n(n+1)(2n+1)6+16×n(n+1)2=n(n+1)6[n(n+1)2+(2n+1)2+12]=n(n+1)6[n2+n+2n+1+12]=n(n+1)6[n2+n+2n+22]=n(n+1)6[n(n+1)+2(n+1)2]=n(n+1)6[(n+1)(n+2)2]=n(n+1)2(n+2)12

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