| Exam Board | OCR |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2006 |
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| Session | January |
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| Topic | Sequences and series, recurrence and convergence |
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5 Use the standard results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } - 6 r ^ { 2 } + 2 r \right) = 2 n ^ { 3 } ( n + 1 )$$