| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2023 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Periodic Sequences |
| Difficulty | Hard +2.3 This AEA question requires deep understanding of periodic sequences through a recurrence relation. Parts (a)-(b) involve algebraic manipulation of fixed points and 2-cycles, but parts (c)-(d) demand significant insight: recognizing that period-2 sequences satisfy the period-4 condition, and deriving the quartic equation for genuine 4-cycles. The multi-layered structure, need for case analysis, and non-standard problem-solving approach place this well above typical A-level questions. |
| Spec | 1.04e Sequences: nth term and recurrence relations4.01a Mathematical induction: construct proofs |
\begin{enumerate}
\item A sequence of non-zero real numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by
\end{enumerate}
$$a _ { n + 1 } = p + \frac { q } { a _ { n } } \quad n \in \mathbb { N }$$
where $p$ and $q$ are real numbers with $q \neq 0$\\
It is known that
\begin{itemize}
\item one of the terms of this sequence is a
\item the sequence is periodic\\
(a) Determine an equation for $q$, in terms of $p$ and $a$, such that the sequence is constant (of period/order one).\\
(b) Determine the value of $p$ that is necessary for the sequence to be of period/order 2.\\
(c) Give an example of a sequence that satisfies the condition in part (b), but is not of period/order 2.\\
(d) Determine an equation for $q$, in terms of $p$ only, such that the sequence has period/order 4.
\end{itemize}
\hfill \mbox{\textit{Edexcel AEA 2023 Q7 [15]}}