| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Sum from n+1 to 2n or similar range |
| Difficulty | Challenging +1.2 Part (a) is routine application of standard summation formulas (Σr, Σr², Σr³) requiring algebraic manipulation but no insight. Part (b) requires the telescoping technique (sum from n to 2n equals sum to 2n minus sum to n-1) and factorization of a quartic expression, which is more challenging than typical but still a standard Further Maths technique with clear methodology. |
| Spec | 1.04g Sigma notation: for sums of series4.06a Summation formulae: sum of r, r^2, r^3 |
\begin{enumerate}
\item (a) Use the standard results for summations to show that, for all positive integers $n$,
\end{enumerate}
$$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 3 r - 1 \right) = \frac { 1 } { 2 } n ( n + 1 ) ^ { 2 } ( n - 2 )$$
(b) Hence show that, for all positive integers $n$,
$$\sum _ { r = n } ^ { 2 n } r \left( 2 r ^ { 2 } - 3 r - 1 \right) = \frac { 1 } { 2 } n ( n - 1 ) ( a n + b ) ( c n + d )$$
where $a$, $b$, $c$ and $d$ are integers to be determined.
\hfill \mbox{\textit{Edexcel F1 2024 Q8 [8]}}