Problem

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

$\sum_{r=1}^{n} (2r - 1)^2 = \frac{1}{3}n(4n^2 - 1)$

for all positive integers $n$.

(b) Hence find the exact value of the sum of the squares of the odd numbers between $200$ and $500$