In this question use the standard results for summations.
Show that for all positive integers \(n\)
$$\sum _ { r = 1 } ^ { n } \left( 12 r ^ { 2 } + 2 r - 3 \right) = A n ^ { 3 } + B n ^ { 2 }$$
where \(A\) and \(B\) are integers to be determined.
Hence determine the value of \(n\) for which
$$\sum _ { r = 1 } ^ { 2 n } r ^ { 3 } - \sum _ { r = 1 } ^ { n } \left( 12 r ^ { 2 } + 2 r - 3 \right) = 270$$