4. Given that
$$( 2 r + 1 ) ^ { 3 } = A r ^ { 3 } + B r ^ { 2 } + C r + 1 ,$$
- find the values of the constants \(A , B\) and \(C\).
- Show that
$$( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 } = 24 r ^ { 2 } + 2$$
- Using the result in part (b) and the method of differences, show that
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$