| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | September |
| Marks | 4 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Simple recurrence evaluation |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring students to discover that the sequence is periodic by computing the first few terms (a₁=3, a₂=-1, a₃=2, a₄=3), then use this 3-cycle pattern to find the sum. While it requires pattern recognition and careful arithmetic rather than just routine application, the periodicity becomes apparent quickly and the calculation is straightforward once discovered. The 'hence' part is trivial addition. This is moderately above average difficulty due to the non-standard recurrence relation and need for insight, but well within reach for FM students. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
A sequence of numbers $a_1, a_2, a_3, ...$ is defined by
$$a_1 = 3$$
$$a_{n+1} = \frac{a_n - 3}{a_n - 2}, \quad n \in \mathbb{N}$$
\begin{enumerate}[label=(\alph*)]
\item Find $\sum_{r=1}^{100} a_r$ [3]
\item Hence find $\sum_{r=1}^{100} a_r + \sum_{r=1}^{99} a_r$ [1]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q2 [4]}}