| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Linear iterative formula u(n+1) = pu(n) + q |
| Difficulty | Moderate -0.3 This is a straightforward recurrence relation question requiring substitution of given terms to find constants, then applying the formula and finding a limit. All steps are routine C2 techniques with no conceptual challenges—slightly easier than average due to the mechanical nature of the algebra and the standard limit method. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(150 = 200p + q\) | M1 | Either equation |
| \(120 = 150p + q\) | A1 | Both (condone embedded values for M1A1) |
| Valid method to solve two simultaneous equations in \(p\) and \(q\) | m1 | To find either \(p\) or \(q\) |
| \(p = 0.6\) | A1 | AG (condone if left as a fraction) |
| \(q = 30\) | B1 | |
| Total (a) | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(u_4 = 102\) | B1F\(\surd\) | Ft on \((72 + q)\) |
| Total (b) | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(L = pL + q\); \(L = 0.6L + 30\) | M1 | |
| \(L = \frac{q}{1-p}\) | m1 | |
| \(L = 75\) | A1F\(\surd\) | Ft on \(2.5q\) |
| Total (c) | 3 |
## Question 5(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $150 = 200p + q$ | M1 | Either equation |
| $120 = 150p + q$ | A1 | Both (condone embedded values for M1A1) |
| Valid method to solve two simultaneous equations in $p$ and $q$ | m1 | To find either $p$ or $q$ |
| $p = 0.6$ | A1 | AG (condone if left as a fraction) |
| $q = 30$ | B1 | |
| **Total (a)** | **5** | |
## Question 5(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $u_4 = 102$ | B1F$\surd$ | Ft on $(72 + q)$ |
| **Total (b)** | **1** | |
## Question 5(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $L = pL + q$; $L = 0.6L + 30$ | M1 | |
| $L = \frac{q}{1-p}$ | m1 | |
| $L = 75$ | A1F$\surd$ | Ft on $2.5q$ |
| **Total (c)** | **3** | |
---5 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 three terms of the sequence are given by
$$u _ { 1 } = 200 \quad u _ { 2 } = 150 \quad u _ { 3 } = 120$$
\begin{enumerate}[label=(\alph*)]
\item Show that $p = 0.6$ and find the value of $q$.
\item Find the value of $u _ { 4 }$.
\item The limit of $u _ { n }$ as $n$ tends to infinity is $L$. Write down an equation for $L$ and hence find the value of $L$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2006 Q5 [9]}}