- (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) 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
$$10 \times 11 + 11 \times 12 + 12 \times 13 + \ldots + 100 \times 101$$