| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Proving standard summation formulae |
| Difficulty | Standard +0.3 This is a guided method-of-differences question with explicit scaffolding through parts (a) and (b). While it requires understanding telescoping series and algebraic manipulation, the structure heavily directs students through each step, making it easier than an average A-level question that requires independent problem-solving or novel insight. |
| Spec | 4.06a Summation formulae: sum of r, r^2, r^34.06b Method of differences: telescoping series |
Given that
$$(2r + 1)^3 = Ar^3 + Br^2 + Cr + 1,$$
\begin{enumerate}[label=(\alph*)]
\item find the values of the constants $A$, $B$ and $C$. [2]
\item Show that
$$(2r + 1)^3 - (2r - 1)^3 = 24r^2 + 2.$$ [2]
\item Using the result in part (b) and the method of differences, show that
$$\sum_{r=1}^n r^2 = \frac{1}{6}n(n + 1)(2n + 1).$$ [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q4 [9]}}