| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find year when threshold exceeded |
| Difficulty | Moderate -0.3 This is a straightforward geometric sequence application with standard techniques: finding the common ratio from given terms, then solving an inequality using logarithms. Part (a) requires solving 30000r³ = 34000 for r, and part (b) requires solving 30000r^N > 75000. While it involves multiple steps and logarithms, these are routine P2 procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks |
|---|---|
| (a) Attempts \(30000 \times r^3 = 34000\) | M1 |
| \(r^3 = \frac{17}{15} \Rightarrow r = 1.0426\) | A1 |
| Hence \(p = 4.26\) | A1ft |
| Answer | Marks |
|---|---|
| (b) Attempts \(30000 \times (1.0426)^N = 75000\) or with \(>\) or \(<\) etc throughout | M1 |
| \((1.0426)^N = \frac{5}{2}\) | A1 |
| Takes logs: \(N = \frac{\log \frac{5}{2}}{\log 1.0426}\) (= awrt 21.96) | M1 |
| \(N = 22\) | A1 |
(a) Attempts $30000 \times r^3 = 34000$ | M1
$r^3 = \frac{17}{15} \Rightarrow r = 1.0426$ | A1
Hence $p = 4.26$ | A1ft
(3 marks)
(b) Attempts $30000 \times (1.0426)^N = 75000$ or with $>$ or $<$ etc throughout | M1
$(1.0426)^N = \frac{5}{2}$ | A1
Takes logs: $N = \frac{\log \frac{5}{2}}{\log 1.0426}$ (= awrt 21.96) | M1
$N = 22$ | A1
(4 marks)
(5 marks)
(8 marks)
---5. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
A colony of bees is being studied.
The number of bees in the colony at the start of the study was 30000
Three years after the start of the study, the number of bees in the colony is 34000
A model predicts that the number of bees in the colony will increase by $p \%$ each year, so that the number of bees in the colony at the end of each year of study forms a geometric sequence.
Assuming the model,
\begin{enumerate}[label=(\alph*)]
\item find the value of $p$, giving your answer to 2 decimal places.
According to the model, at the end of $N$ years of study the number of bees in the colony exceeds 75000
\item Find, showing all steps in your working, the smallest integer value of $N$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2020 Q5 [8]}}