Problem

In this question use the standard results for summations.

(a) Show that for all positive integers $n$

$\sum_{r=1}^n (12r^2 + 2r - 3) = An^3 + Bn^2$

where $A$ and $B$ are integers to be determined.

(b) Hence determine the value of $n$ for which

$\sum_{r=1}^{2n} r^3 - \sum_{r=1}^n (12r^2 + 2r - 3) = 270$