- (a) Prove by induction that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( 2 r + 1 ) = \frac { 1 } { 2 } n ( n + 1 ) ^ { 2 } ( n + 2 )$$
(b) Hence, show that, for all positive integers \(n\),
$$\sum _ { r = n } ^ { 2 n } r ( r + 1 ) ( 2 r + 1 ) = \frac { 1 } { 2 } n ( n + 1 ) ( a n + b ) ( c n + d )$$
where \(a\), \(b\), \(c\) and \(d\) are integers to be determined.