Problem

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

$\sum_{r=1}^{n} r(r-1)(r-3) = \frac{1}{12}n(n+1)(n-1)(3n-10)$

(b) Hence show that

$\sum_{r=n+1}^{2n+1} r(r-1)(r-3) = \frac{1}{12}n(n+1)(an^2 + bn + c)$

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