| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Finding constants from given sum formula |
| Difficulty | Moderate -0.3 This is a straightforward algebraic manipulation question requiring standard sum formulas (∑r and ∑r³) that Further Maths students are expected to know. The task is purely mechanical: expand the left side, substitute known formulas, simplify to a quartic in n, then factor to match the given form and identify constants. No problem-solving insight or novel techniques required, making it slightly easier than average despite being Further Maths content. |
| Spec | 4.06a Summation formulae: sum of r, r^2, r^3 |
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\item Use the standard results for $\sum _ { r = 1 } ^ { n } r$ and for $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to show that, for all positive integers $n$,
\end{enumerate}
$$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { n } { 4 } ( n + a ) ( n + b ) ( n + c )$$
where $a$, $b$ and $c$ are integers to be found.\\
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\hfill \mbox{\textit{Edexcel F1 2018 Q1 [4]}}