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Question

# What is the sum of 11+22+33+...n2?

A
n(n1)(2n+1)6
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B
n(n+1)(n+1)6
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C
n(n+1)(2n+1)6
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D
n(n+1)(2n+1)3
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Solution

## The correct option is C n(n+1)(2n+1)6Here Sn=11+22+33+...n2We consider the identity x3−(x−1)3=3x2−3x+1When x = 1, 13−(1−1)3=3(1)2−3(1)+1When x = 2, 23−(2−1)3=3(2)2−3(2)+1When x = 3, 33−(3−1)3=3(3)2−3(3)+1.......n3−(n−1)3=3n2−3n+1Adding vertically both sides, n3=3(12+22+32+...n2)−3(1+2+3...+n)+(1+1+1..n)3(12+22+32+...n2)=n3+3n(n+1)2−n= 2n3+3n2+n2= n(2n2+3n+1)2∑n2=n(n+1)(2n+1)6

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