| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2022 |
| Session | October |
| Marks | 7 |
| Topic | Sequences and Series |
| Type | Periodic Sequences |
| Difficulty | Standard +0.8 This question requires students to compute terms of a recurrence relation, identify a periodic pattern (the sequence cycles with period 3), and use this pattern to find a specific term and sum. While the individual calculations are straightforward, recognizing the periodicity and applying it to find uāā and the sum over 99 terms requires problem-solving insight beyond routine exercises. The multi-step reasoning and pattern recognition elevate this above average difficulty. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04f Sequence types: increasing, decreasing, periodic |
A sequence is defined by
$$u_1 = 3$$
$$u_{n+1} = 2 - \frac{4}{u_n}, \quad n \geq 1$$
Find the exact values of
\begin{enumerate}[label=(\alph*)]
\item $u_2$, $u_3$ and $u_4$ [3]
\item $u_{61}$ [1]
\item $\sum_{i=1}^{99} u_i$ [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2022 Q7 [7]}}