| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 6 |
| 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 solving a quadratic equation. Part (a) is immediate substitution, part (b) is a simple 'show that' requiring one more substitution, and part (c) involves solving a basic quadratic. No problem-solving insight needed—purely mechanical application of given formulas. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.04e Sequences: nth term and recurrence relations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (a) \(x_2 = a - 3\) | B1 (1) | B1: for \(a\times1-3\) or better; give for \(a-3\) in part (a) or if it appears in (b) they must state \(x_2=a-3\) |
| (b) \(x_3 = ax_2 - 3\) or \(a(a-3)-3\) | M1 | M1: clear show that; usually \(a(a-3)-3\) |
| \(= a(a-3)-3 = a^2-3a-3\) | A1cso (2) | A1: correct processing to printed answer; both lines needed; no incorrect working seen |
| (c) \(a^2-3a-3=7\) | M1 | 1st M1: attempt to form correct equation and collect terms; must be quadratic |
| \(a^2-3a-10=0\) | ||
| \((a-5)(a+2)=0\) | dM1 | 2nd dM1 (dep on 1st M1): attempt to factorise/solve their 3TQ=0 |
| \(a=5\) or \(a=-2\) | A1 (3) | A1: both correct answers; allow \(x=\ldots\); give 3/3 for correct answers with no working |
# Question 5:
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(a)** $x_2 = a - 3$ | B1 (1) | B1: for $a\times1-3$ or better; give for $a-3$ in part (a) or if it appears in (b) they must state $x_2=a-3$ |
| **(b)** $x_3 = ax_2 - 3$ or $a(a-3)-3$ | M1 | M1: clear show that; usually $a(a-3)-3$ |
| $= a(a-3)-3 = a^2-3a-3$ | A1cso (2) | A1: correct processing to printed answer; both lines needed; no incorrect working seen |
| **(c)** $a^2-3a-3=7$ | M1 | 1st M1: attempt to form correct equation and collect terms; must be quadratic |
| $a^2-3a-10=0$ | | |
| $(a-5)(a+2)=0$ | dM1 | 2nd dM1 (dep on 1st M1): attempt to factorise/solve their 3TQ=0 |
| $a=5$ or $a=-2$ | A1 (3) | A1: both correct answers; allow $x=\ldots$; give 3/3 for correct answers with no working |5. A sequence $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ is defined by
$$\begin{gathered}
x _ { 1 } = 1 , \\
x _ { n + 1 } = a x _ { n } - 3 , n \geqslant 1 ,
\end{gathered}$$
where $a$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $x _ { 2 }$ in terms of $a$.
\item Show that $x _ { 3 } = a ^ { 2 } - 3 a - 3$.
Given that $x _ { 3 } = 7$,
\item find the possible values of $a$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2008 Q5 [6]}}