| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: find parameter from given term |
| Difficulty | Moderate -0.8 This is a straightforward recurrence relation question requiring only direct substitution and basic algebraic manipulation. Parts (a)-(c) involve simple substitution and solving a quadratic equation, while part (d) requires recognizing that the sequence becomes constant (x₂=1 implies all subsequent terms equal 1). No novel insight needed—purely mechanical application of the given formula. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(l(p+1)\) or \(p+1\) | B1 | |
| (b) \(((a))(p + (a))\) ([(a) must be a function of p].) | M1 | \([(p+1)(p+p+1)]\) |
| \(= 1 + 3p + 2p^2\) | A1also |
| Answer | Marks | Guidance |
|---|---|---|
| (c) \(1 + 3p + 2p^2 = 1\) | M1 | |
| \(p(2p + 3) = 0\) | M1 | |
| \(p = ...\) | M1 | |
| \(p = -\frac{3}{2}\) | A1 | (ignore \(p = 0\), if seen, even if 'chosen' as the answer) |
| Answer | Marks | Guidance |
|---|---|---|
| (d) Noting that even terms are the same. | M1 | This M mark can be implied by listing at least 4 terms, e.g. \(1, -\frac{1}{2}, 1, -\frac{1}{2}, ...\) |
| \(x_{2008} = -\frac{1}{2}\) | A1 |
(a) $l(p+1)$ or $p+1$ | B1 |
(b) $((a))(p + (a))$ ([(a) must be a function of p].) | M1 | $[(p+1)(p+p+1)]$
$= 1 + 3p + 2p^2$ | A1also |
**Total for (b): 2 marks**
(c) $1 + 3p + 2p^2 = 1$ | M1 |
$p(2p + 3) = 0$ | M1 |
$p = ...$ | M1 |
$p = -\frac{3}{2}$ | A1 | (ignore $p = 0$, if seen, even if 'chosen' as the answer)
**Total for (c): 3 marks**
(d) Noting that even terms are the same. | M1 | This M mark can be implied by listing at least 4 terms, e.g. $1, -\frac{1}{2}, 1, -\frac{1}{2}, ...$
$x_{2008} = -\frac{1}{2}$ | A1 |
**Total for (d): 2 marks**
**Total: 8 marks**
---\begin{enumerate}
\item A sequence is given by:
\end{enumerate}
$$\begin{aligned}
& x _ { 1 } = 1 \\
& x _ { n + 1 } = x _ { n } \left( p + x _ { n } \right)
\end{aligned}$$
where $p$ is a constant ( $p \neq 0$ ) .\\
(a) Find $x _ { 2 }$ in terms of $p$.\\
(b) Show that $x _ { 3 } = 1 + 3 p + 2 p ^ { 2 }$.
Given that $x _ { 3 } = 1$,\\
(c) find the value of $p$,\\
(d) write down the value of $x _ { 2008 }$.\\
\hfill \mbox{\textit{Edexcel C1 2008 Q7 [8]}}