5. (a) Use the results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), to prove that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 5 ) = \frac { 1 } { 4 } n ( n + 1 ) ( n + 2 ) ( n + 7 )$$ for all positive integers \(n\).
(b) Hence, or otherwise, find the value of $$\sum _ { r = 20 } ^ { 50 } r ( r + 1 ) ( r + 5 )$$