Moderate -0.5 This is a straightforward algebraic manipulation question requiring recall of two standard summation formulae and basic algebra to verify the result. While it's from Further Maths FP1, it requires no proof technique, no induction despite the topic label, and no problem-solving—just direct substitution and simplification. It's easier than average but not trivial since it involves manipulating summation notation correctly.
3 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 1 \right) = n ^ { 3 }$$
3 Use the standard results for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ to show that, for all positive integers $n$,
$$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 1 \right) = n ^ { 3 }$$
\hfill \mbox{\textit{OCR FP1 2007 Q3 [6]}}