| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Linear iterative formula u(n+1) = pu(n) + q |
| Difficulty | Standard +0.3 This is a straightforward recurrence relation problem requiring substitution to form simultaneous equations, then solving a quadratic. Part (a) involves algebraic manipulation with 5 marks suggesting multiple steps but standard technique. Part (b) tests understanding of limits in recurrence relations (L = pL + q), which is a common textbook exercise. Slightly easier than average due to routine methods throughout. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
A sequence is defined by
$$u_1 = 600$$
$$u_{n+1} = pu_n + q$$
where $p$ and $q$ are constants.
It is given that $u_2 = 500$ and $u_4 = 356$
\begin{enumerate}[label=(\alph*)]
\item Find the two possible values of $u_3$ [5 marks]
\item When $u_n$ is a decreasing sequence, the limit of $u_n$ as $n$ tends to infinity is $L$. Write down an equation for $L$ and hence find the value of $L$. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2022 Q11 [7]}}