2 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
$$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$
2 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
$$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$
\hfill \mbox{\textit{CAIE FP1 2014 Q2}}