9. (a) Prove by induction that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ Using the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\),
(b) show that $$\sum _ { r = 1 } ^ { n } ( r + 2 ) ( r + 3 ) = \frac { 1 } { 3 } n \left( n ^ { 2 } + a n + b \right) ,$$ where \(a\) and \(b\) are integers to be found.
(c) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 2 ) ( r + 3 ) = \frac { 1 } { 3 } n \left( 7 n ^ { 2 } + 27 n + 26 \right)$$