| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2026 |
| Session | November |
| Marks | 5 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Partial fractions then method of differences |
| Difficulty | Standard +0.8 This question requires partial fraction decomposition of 4/(r²-1), applying the method of differences to identify telescoping terms, and algebraic manipulation to match the given form and determine constants p and q. While the technique is standard for Further Maths, it demands careful execution across multiple steps and algebraic insight to recognize the telescoping pattern, placing it moderately above average difficulty. |
| Spec | 4.06b Method of differences: telescoping series |
Use the method of differences to prove that for $n > 2$
$$\sum_{r=2}^{n} \frac{4}{r^2-1} = \frac{(pn+q)(n-1)}{n(n+1)}$$
where $p$ and $q$ are constants to be determined. [5]
\hfill \mbox{\textit{SPS SPS FM Pure 2026 Q5 [5]}}