| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Modelling with Recurrence Relations |
| Difficulty | Easy -1.2 This is a straightforward recurrence relation question requiring only direct substitution and basic arithmetic for parts (a) and (b), with part (c) being a simple equilibrium condition (u_{n+1} = u_n). All parts are routine C1-level exercises with no problem-solving insight needed—just mechanical application of the given formula. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
Initially the number of fish in a lake is 500 000. The population is then modelled by the recurrence relation
$$u_{n+1} = 1.05u_n - d, \quad u_0 = 500000.$$
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. [3]
\end{enumerate}
Given that $d = 100000$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item show that the population of fish dies out during the sixth year. [3]
\item Find the value of $d$ which would leave the population each year unchanged. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q12 [8]}}