| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Compound growth applications |
| Difficulty | Moderate -0.3 This is a straightforward application of geometric sequences with real-world context. Part (a) requires finding the common ratio and using the nth term formula, part (b) involves the sum formula for a GP, and part (c) requires solving an inequality. All techniques are standard C2 content with no novel insight required, though the multi-part structure and time conversions add minor complexity beyond the most basic GP questions. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(r = \frac{114}{120} = 0.95\) | M1 | |
| \(u_5 = 120 \times (0.95)^4 = 97.74\) | M1 | |
| \(\therefore\) 1 hour 38 minutes | A1 | |
| (b) \(S_\infty = \frac{120[1-(0.95)^8]}{1-0.95}\) | M1 A1 | |
| \(= 807.79\ldots\) minutes \(= 13\) hours 28 minutes | A1 | |
| (c) \(120 \times (0.95)^{n-1} < 60\) | M1 | |
| \((n-1) \lg 0.95 < \lg 0.5\) | M1 | |
| \(n > \frac{\lg 0.5}{\lg 0.95} + 1\) | A1 | |
| \(n > 14.51 \quad \therefore 15\) papers | A1 | (10 marks) |
**(a)** $r = \frac{114}{120} = 0.95$ | M1 |
$u_5 = 120 \times (0.95)^4 = 97.74$ | M1 |
$\therefore$ 1 hour 38 minutes | A1 |
**(b)** $S_\infty = \frac{120[1-(0.95)^8]}{1-0.95}$ | M1 A1 |
$= 807.79\ldots$ minutes $= 13$ hours 28 minutes | A1 |
**(c)** $120 \times (0.95)^{n-1} < 60$ | M1 |
$(n-1) \lg 0.95 < \lg 0.5$ | M1 |
$n > \frac{\lg 0.5}{\lg 0.95} + 1$ | A1 |
$n > 14.51 \quad \therefore 15$ papers | A1 | **(10 marks)**7. A student completes a mathematics course and begins to work through past exam papers. He completes the first paper in 2 hours and the second in 1 hour 54 minutes.
Assuming that the times he takes to complete successive papers form a geometric sequence,
\begin{enumerate}[label=(\alph*)]
\item find, to the nearest minute, how long he will take to complete the fifth paper,
\item show that the total time he takes to complete the first eight papers is approximately 13 hours 28 minutes,
\item find the least number of papers he must work through if he is to complete a paper in less than one hour.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q7 [10]}}