Problem

3. (a) Use the standard results for $\displaystyle\sum_{r=1}^{n} r^2$ and $\displaystyle\sum_{r=1}^{n} r^3$ to show that for all positive integers $n$

$\sum_{r=1}^{n} r^2(2r + 3) = \frac{1}{2}n^2(n + 1)(n + 3n + 1)$

(b) Hence calculate the value of $\displaystyle\sum_{r=10}^{25} r^2(2r + 3)$