| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | September |
| Marks | 5 |
| Topic | Geometric Sequences and Series |
| Type | Recursive sequence definition |
| Difficulty | Standard +0.3 This question tests understanding of recursive sequences and limits, requiring students to use the limit condition L = pL + q to find p, then apply the recurrence relation. While it involves multiple steps and the concept of limits at infinity, the algebraic manipulation is straightforward and the problem follows a standard template for this topic type. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04f Sequence types: increasing, decreasing, periodic |
7. The $n$th term of a sequence is $u _ { n }$. The sequence is defined by
$$u _ { n + 1 } = p u _ { n } + q$$
where $p$ and $q$ are constants.\\
The first two terms of the sequence are given by $u _ { 1 } = 96$ and $u _ { 2 } = 72$.\\
The limit of $u _ { n }$ as $n$ tends to infinity is 24 .
\begin{enumerate}[label=(\alph*)]
\item Show that $p = \frac { 2 } { 3 }$.
\item Find the value of $u _ { 3 }$.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q7 [5]}}