- (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { n } { 3 } \left( a n ^ { 2 } - 1 \right)$$
where \(a\) is a constant to be determined.
(b) Hence determine the sum of the squares of all positive odd three-digit integers.