Show that the equation \(4 x ^ { 3 } - x - 540000 = 0\) has a root, \(\alpha\), in the interval \(51 < \alpha < 52\).
It is given that \(S _ { n } = \sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 }\).
Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that \(S _ { n } = \frac { n } { 3 } \left( k n ^ { 2 } - 1 \right)\), where \(k\) is an integer to be found.
Hence show that \(6 S _ { n }\) can be written as the product of three consecutive integers.
Find the smallest value of \(N\) for which the sum of the squares of the first \(N\) odd numbers is greater than 180000 .