Problem

(a) Use the standard results for summations to show that for all positive integers $n$

$\sum_{r=1}^n r(r+1)(r+5) = \frac{1}{4}n(n+a)(n+b)(n+c)$

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

(b) Hence determine the value of

$20 \times 21 \times 25 + 21 \times 22 \times 26 + \dots + 40 \times 41 \times 45$