| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | November |
| Marks | 7 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Partial fractions then method of differences |
| Difficulty | Standard +0.8 This is a Further Maths question requiring partial fractions with a non-standard decomposition, followed by method of differences with telescoping series. While the techniques are standard for FM students, the specific partial fraction form requires insight (recognizing the grouping structure), and the telescoping requires careful tracking of which terms survive. The calculation is multi-step but methodical once the approach is identified. |
| Spec | 4.06b Method of differences: telescoping series |
In this question you must show detailed reasoning.
(a) Given that
$$\frac{1}{r(r + 1)(r + 2)} = \frac{A}{r(r + 1)} + \frac{B}{(r + 1)(r + 2)}$$
show that $A = \frac{1}{2}$ and find the value of $B$. [3]
(b) Use the method of differences to find
$$\sum_{r=10}^{98} \frac{1}{r(r + 1)(r + 2)}$$
giving your answer as a rational number. [4]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q4 [7]}}