Expand and simplify \((r+1)^{4}-r^{4}\).

Use the method of differences together with the standard results for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^{2}\) to show that
\(\sum_{r=1}^{n} r^{3}=\frac{1}{4} n^{2}(n+1)^{2} .\)