| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2022 |
| Session | October |
| Marks | 7 |
| Topic | Sequences and Series |
| Type | Periodic Sequences |
| Difficulty | Challenging +1.2 This is a multi-part recurrence relation problem requiring algebraic manipulation to find k, then exploiting periodicity to sum terms. Part (a) requires computing a₂ and a₃, then using the period-3 condition (a₄ = a₁) to derive the quadratic—this is moderately challenging but systematic. Parts (b) and (c) are straightforward once k is found. The problem requires careful algebra and understanding of periodic sequences, placing it above average difficulty but not requiring exceptional insight. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04g Sigma notation: for sums of series |
A sequence of numbers $a_1, a_2, a_3, \ldots$ is defined by
$$a_{n+1} = \frac{k(a_n + 2)}{a_n}$$, $n \in \mathbb{N}$
where $k$ is a constant.
Given that
\begin{itemize}
\item the sequence is a periodic sequence of order 3
\item $a_1 = 2$
\end{itemize}
\begin{enumerate}[label=(\alph*)]
\item show that
$$k^2 + k - 2 = 0$$ [3]
\item For this sequence explain why $k \neq 1$ [1]
\item Find the value of
$$\sum_{r=1}^{80} a_r$$ [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2022 Q9 [7]}}