9. (a) Prove by induction that, for \(n \in \mathbb { N }\)
$$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$
(b) Using the standard summation formulae, show that
$$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = \frac { 1 } { 4 } n ( n + A ) ( n + B ) ( n + C )$$
where \(A , B\) and \(C\) are constants to be determined.
(c) Determine the value of \(n\) for which
$$3 \sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = 17 \sum _ { r = n } ^ { 2 n } r ^ { 2 }$$
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