Problem

7. (a) Prove by induction that for $n \in \mathbb{Z}^+$

$\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1)$

(b) Hence show that

$\sum_{r=1}^{n} (r+2)^2 = \frac{1}{6}(an^3 + bn^2 + cn)$

where $a$, $b$ and $c$ are integers to be found.

(c) Using your answers to part (b), find the value of

$\sum_{r=10}^{25} (r+2)^2$