Problem

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

$\sum_{r=0}^{n} (r+1)(r+2) = \frac{1}{3}(n+1)(n+2)(n+3)$

(b) Hence determine the value of

$10 \times 11 + 11 \times 12 + 12 \times 13 + \ldots + 100 \times 101$