Problem

1. Use the standard results for $\displaystyle\sum_{r=1}^{n} r^2$ and $\displaystyle\sum_{r=1}^{n} r^3$ to show that, for all positive integers $n$

$\sum_{r=1}^{n} r^2(r+2) = \frac{1}{12}n(n+1)(an^2 + bn + c)$

where $a$, $b$ and $c$ are integers to be determined.