- (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 8 \right) = \frac { 1 } { 3 } n ( n - a ) ( n + a )$$
where \(a\) is a positive integer to be determined.
(b) Hence, or otherwise, state the positive value of \(n\) that satisfies
$$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 8 \right) = 0$$
Given that
$$\sum _ { r = 3 } ^ { 17 } \left( k r ^ { 3 } + r ^ { 2 } - r - 8 \right) = 6710 \quad \text { where } k \text { is a constant }$$
(c) find the exact value of \(k\).