- (a) Given that \(u _ { n + 1 } = 5 u _ { n } + 4 , u _ { 1 } = 4\), prove by induction that \(u _ { n } = 5 ^ { n } - 1\).
(b) For all positive integers, \(n \geq 2\), prove by induction that
$$\sum _ { r = 2 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { 1 } { 12 } n ( n - 1 ) ( n + 1 ) ( 3 n + 2 )$$