| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: find specific terms |
| Difficulty | Moderate -0.3 This is a straightforward recurrence relation question requiring simple iterative calculations and basic interpretation. Part (a) involves direct substitution (3 iterations), part (b) requires continuing iterations until the population becomes negative, and part (c) involves solving a simple linear equation (setting u_{n+1} = u_n). While it has multiple parts and requires understanding of recurrence relations, all steps are mechanical with no conceptual challenges beyond applying the given formula repeatedly. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
| Answer | Marks | Guidance |
|---|---|---|
| \(u_1 = 1.05 \times 500000 - 15000 = 510000\) | M1 | |
| \(u_2 = 520500\) | ||
| \(u_3 = 531525\) | A1 | (all 3) |
| The population is increasing | B1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(u_1 = 425000\), \(u_2 = 346250\), \(u_3 = 263562.5\), \(u_4 = 176740.625\) | M1 | |
| \(u_5 = 85577.64\ldots\) | A1 | |
| \(u_6 = -10143.46\ldots\) | ||
| \(u_5 > 0\), \(u_6 < 0\) so population died out during 6th year | B1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Require \(u_1 = u_0\), i.e. \(1.05 \times 500000 - d = 500000\) | M1 | |
| i.e. \(d = 0.05 \times 500000\) | ||
| i.e. \(d = 25000\) | A1 | (2 marks) |
## Question 6:
### Part (a)
$u_1 = 1.05 \times 500000 - 15000 = 510000$ | M1 |
$u_2 = 520500$ | |
$u_3 = 531525$ | A1 | (all 3)
The population is increasing | B1 | (3 marks)
### Part (b)
$u_1 = 425000$, $u_2 = 346250$, $u_3 = 263562.5$, $u_4 = 176740.625$ | M1 |
$u_5 = 85577.64\ldots$ | A1 |
$u_6 = -10143.46\ldots$ | |
$u_5 > 0$, $u_6 < 0$ so population died out during 6th year | B1 | (3 marks)
### Part (c)
Require $u_1 = u_0$, i.e. $1.05 \times 500000 - d = 500000$ | M1 |
i.e. $d = 0.05 \times 500000$ | |
i.e. $d = 25000$ | A1 | (2 marks)
---6. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation $\quad u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000$.\\
In this relation $u _ { n }$ is the number of fish in the lake after $n$ years and $d$ is the number of fish which are caught each year.\\
Given that $d = 15000$,
\begin{enumerate}[label=(\alph*)]
\item calculate $u _ { 1 } , u _ { 2 }$ and $u _ { 3 }$ and comment briefly on your results.
Given that $d = 100000$,
\item show that the population of fish dies out during the sixth year.
\item Find the value of $d$ which would leave the population each year unchanged.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q6 [8]}}