| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Partial fractions then method of differences |
| Difficulty | Standard +0.3 This is a standard telescoping series question requiring partial fractions (a routine A-level technique) followed by straightforward algebraic manipulation. Part (a) is a guided proof with a clear method suggested, and part (b) is simple arithmetic using the result from (a). While it requires multiple steps, the techniques are all standard FP2 content with no novel insight needed. |
| Spec | 1.02y Partial fractions: decompose rational functions4.06b Method of differences: telescoping series |
\begin{enumerate}[label=(\alph*)]
\item By expressing $\frac{2}{4r^2 - 1}$ in partial fractions, or otherwise, prove that
$$\sum_{r=1}^{n} \frac{2}{4r^2 - 1} = 1 - \frac{1}{2n + 1}.$$ [3]
\item Hence find the exact value of $\sum_{r=11}^{20} \frac{2}{4r^2 - 1}$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q41 [5]}}