Moderate -0.3 This is a straightforward algebraic manipulation question requiring expansion of r(r-1)(r+1) to r³-r, then applying two standard summation formulae and factorising. It's slightly easier than average as it's a direct application of given formulae with minimal problem-solving, though the algebraic manipulation and factorisation require some care.
3 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to find
$$\sum _ { r = 1 } ^ { n } r ( r - 1 ) ( r + 1 ) ,$$
expressing your answer in a fully factorised form.
3 Use the standard results for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to find
$$\sum _ { r = 1 } ^ { n } r ( r - 1 ) ( r + 1 ) ,$$
expressing your answer in a fully factorised form.
\hfill \mbox{\textit{OCR FP1 2007 Q3 [6]}}