| Exam Board | SPS |
|---|---|
| Module | SPS SM Mechanics (SPS SM Mechanics) |
| Year | 2022 |
| Session | February |
| Marks | 6 |
| Topic | Fixed Point Iteration |
| Type | Apply iteration to find root (pure fixed point) |
| Difficulty | Standard +0.3 This is a straightforward recurrence relation problem requiring algebraic substitution and solving a quadratic. Part (a) involves computing u₂ and u₃ in terms of k, then substituting into the given condition—mechanical but with several steps. Part (b) solves the quadratic and selects the appropriate root (likely using convergence or integer constraints). Part (c) is direct substitution. The question is slightly easier than average as it's highly structured with clear signposting and uses standard A-level techniques without requiring novel insight. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
The sequence $u_1, u_2, u_3, \ldots$ is defined by
$$u_{n+1} = k - \frac{24}{u_n} \quad u_1 = 2$$
where $k$ is an integer.
Given that $u_1 + 2u_2 + u_3 = 0$
\begin{enumerate}[label=(\alph*)]
\item show that
$$3k^2 - 58k + 240 = 0$$
[3]
\item Find the value of $k$, giving a reason for your answer.
[2]
\item Find the value of $u_3$
[1]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Mechanics 2022 Q3 [6]}}