| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | First-Order Linear Recurrence Relations |
| Difficulty | Standard +0.3 This is a straightforward application of first-order linear recurrence relations. Part (a) uses the standard limit condition (L = pL + q) combined with one given term to find p, requiring only algebraic manipulation. Part (b) is direct substitution. The question is slightly above average difficulty due to the limit concept, but the method is standard and well-practiced in C2. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
| Answer | Marks | Guidance |
|---|---|---|
| \(72 = 96p + q\) | M1 | |
| \(24 = 24p + q\) | M1 | OE |
| \(48 = 72p\) | m1 | Valid method to solve the correct two simultaneous eqns in \(p\) and \(q\) to at least the stage \(48 = 72p\) OE |
| \(p = \left(\frac{48}{72}\right) = \frac{2}{3}\) | A1 | AG CSO |
| Answer | Marks | Guidance |
|---|---|---|
| \(q = 8\) | B1 | Award if seen at any stage in Q7 |
| \(u_3 = 48 + q\) (\(u_3 = 56\)) | B1F | If not 56, ft on (48 + c's \(q\)) provided at least M1 scored in part (a). |
| Total | 6 |
**7(a)**
$72 = 96p + q$ | M1 |
$24 = 24p + q$ | M1 | OE
$48 = 72p$ | m1 | Valid method to solve the correct two simultaneous eqns in $p$ and $q$ to at least the stage $48 = 72p$ OE
$p = \left(\frac{48}{72}\right) = \frac{2}{3}$ | A1 | AG CSO | 4
**7(b)**
$q = 8$ | B1 | Award if seen at any stage in Q7
$u_3 = 48 + q$ ($u_3 = 56$) | B1F | If not 56, ft on (48 + c's $q$) provided at least M1 scored in part (a). | 2
Total | | 6
---7 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 two terms of the sequence are given by $u _ { 1 } = 96$ and $u _ { 2 } = 72$.\\
The limit of $u _ { n }$ as $n$ tends to infinity is 24 .
\begin{enumerate}[label=(\alph*)]
\item Show that $p = \frac { 2 } { 3 }$.
\item Find the value of $u _ { 3 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2013 Q7 [6]}}