| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Method of differences with given identity |
| Difficulty | Standard +0.3 Part (a) is a routine algebraic verification requiring combining fractions over a common denominator. Part (b) is a standard method of differences application with telescoping series—a technique commonly practiced in A-level Further Maths. The question is straightforward once the telescoping pattern is recognized, requiring no novel insight, making it slightly easier than average. |
| Spec | 4.06b Method of differences: telescoping series |
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac{1}{9r - 4} - \frac{1}{9r + 5} = \frac{9}{(9r - 4)(9r + 5)}$$
[2 marks]
\item Hence use the method of differences to find
$$\sum_{r=1}^{n} \frac{1}{(9r - 4)(9r + 5)}.$$
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2021 Q9 [7]}}