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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