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Saturday, December 4, 2021

2021/106) Evaluate closed form of 1^2+2^2+3^2+\cdots+n^2

We have

(k+1)^3 = k^ 3 + 3k ^2 + 3k + 1

Or (k+1)^3 - k^ 3 = 3k ^2 + 3k + 1

Adding from 1 to n we get

\sum_{k=1}^n((k+1)^3 - k^ 3) = 3\sum_{k=1}^n k^2 + 3\sum_{k=1}^nk + \sum_{k=1}^n1

The LHS is a telespcopic sum = (n+1)^3-1

We know \sum_{k=1}^nk = \frac{n(n+1)}{2}

so we get (n+1)^3 - 1 = 3\sum_{k=1}^n k^2 + 3\frac{n(n+1)}{2} + n

or  3\sum_{k=1}^n k^2 = (n+1)^3 - 1 -  3\frac{n(n+1)}{2} - n

=n^3 + 3n^2 + 3n + 1 - 3\frac{n^2+n}{2} - n
= \frac{1}{2}(2n^3 + 6n^2 + 6n - 3n^2 -3n -n )
= \frac{1}{2}(2n^3 + 3n^2 + 2n )
=\frac{1}{2}n(2n^2+ 3n + 2)
= \frac{1}{2}n(n+2)(2n+1)

so \sum_{k=1}^n k^2 = \frac{1}{6}(n(n+2)(2n+1)

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