Problem

(a) Using the formulae for $\displaystyle\sum_{r=1}^{n} r$ and $\displaystyle\sum_{r=1}^{n} r^2$, show that

$\sum_{r=1}^{n} (r + 1)(r + 5) = \frac{1}{6}n(n + 7)(2n + 7)$

for all positive integers $n$.

(b) Hence show that

$\sum_{r=n+1}^{2n} (r + 1)(r + 5) = \frac{7}{6}n(n + 1)(an + b)$

where $a$ and $b$ are integers to be determined.