| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2022 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Sum from n+1 to 2n or similar range |
| Difficulty | Standard +0.8 This is a substantial Further Maths question requiring proof by induction, algebraic manipulation of summation formulae, and solving a complex equation involving two different summations. While the induction proof is standard, part (c) requires non-trivial manipulation to isolate n and involves working with summations over different ranges, making it moderately challenging even for Further Maths students. |
| Spec | 4.01a Mathematical induction: construct proofs4.06a Summation formulae: sum of r, r^2, r^3 |
\begin{enumerate}[label=(\alph*)]
\item Prove by induction that, for $n \in \mathbb{N}$
$$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2$$
[5]
\item Using the standard summation formulae, show that
$$\sum_{r=1}^{n} r(r+1)(r-1) = \frac{1}{4}n(n+A)(n+B)(n+C)$$
where $A$, $B$ and $C$ are constants to be determined.
[4]
\item Determine the value of $n$ for which
$$3\sum_{r=1}^{n} r(r+1)(r-1) = 17\sum_{r=n}^{2n} r^2$$
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2022 Q9 [14]}}