| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| 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 | Standard +0.3 This is a standard linear recurrence relation problem requiring algebraic manipulation of three equations in three unknowns. Students must use the convergence condition (L = pL + q) and two term values to solve simultaneously. While it involves multiple steps, the techniques are routine for C2 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04f Sequence types: increasing, decreasing, periodic |
| Answer | Marks | Guidance |
|---|---|---|
| \(u_3 = pu_2 + q\): \(132 = 160p + q\) | M1 | Setting up equation using terms |
| Limit \(L = 20\): \(20 = 20p + q\) | M1 | Using limit condition \(L = pL + q\) |
| Subtracting: \(112 = 140p\), so \(p = \frac{4}{5}\) (or \(0.8\)) | A1 | |
| \(q = 20 - 16 = 4\) | A1 A1 |
| Answer | Marks |
|---|---|
| \(160 = \frac{4}{5}u_1 + 4\) | M1 |
| \(u_1 = 195\) | A1 |
## Question 5:
**Part (a):** Find $p$ and $q$
| $u_3 = pu_2 + q$: $132 = 160p + q$ | M1 | Setting up equation using terms |
| Limit $L = 20$: $20 = 20p + q$ | M1 | Using limit condition $L = pL + q$ |
| Subtracting: $112 = 140p$, so $p = \frac{4}{5}$ (or $0.8$) | A1 | |
| $q = 20 - 16 = 4$ | A1 A1 | |
**Part (b):** Find the first term
| $160 = \frac{4}{5}u_1 + 4$ | M1 | |
| $u_1 = 195$ | A1 | |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 second term of the sequence is 160 . The third term of the sequence is 132 .\\
The limit of $u _ { n }$ as $n$ tends to infinity is 20 .
\begin{enumerate}[label=(\alph*)]
\item Find the value of $p$ and the value of $q$.
\item Hence find the value of the first term of the sequence.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2015 Q5 [6]}}