| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find sum to infinity |
| Difficulty | Moderate -0.8 This is a straightforward application of standard geometric series formulas with no conceptual challenges. Part (i) uses the sum to infinity formula directly, part (ii) uses the finite sum formula, and part (iii) requires basic logarithm manipulation but follows a routine method. All three parts are textbook exercises requiring only formula recall and arithmetic. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<11.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| \(S_\infty = \frac{20}{1-0.9} = 200\) | M1, A1 2 | Attempt use of \(S_\infty = \frac{a}{1-r}\); obtain 200 |
| Answer | Marks | Guidance |
|---|---|---|
| \(S_{30} = \frac{20(1-0.9^{30})}{1-0.9} = 192\) | M1, A1 2 | Attempt use of correct sum formula for GP with \(n=30\); obtain 192 or better |
| Answer | Marks | Guidance |
|---|---|---|
| \(20 \times 0.9^{p-1} < 0.4\), i.e. \(0.9^{p-1} < 0.02\) | B1 | Correct \(20 \times 0.9^{p-1}\) seen or implied |
| \((p-1)\log 0.9 < \log 0.02\) | M1 | Link to 0.4, rearrange to \(0.9^k = c\), introduce logarithms, drop power, or equiv correct method |
| \(p - 1 > \frac{\log 0.02}{\log 0.9}\) | M1 | Correct method for solving their (in)equation |
| \(p > 38.1\), hence \(p = 39\) | A1 4 | State 39 (not inequality), no wrong working seen |
# Question 6:
## Part (i):
| $S_\infty = \frac{20}{1-0.9} = 200$ | M1, A1 **2** | Attempt use of $S_\infty = \frac{a}{1-r}$; obtain 200 |
|---|---|---|
## Part (ii):
| $S_{30} = \frac{20(1-0.9^{30})}{1-0.9} = 192$ | M1, A1 **2** | Attempt use of correct sum formula for GP with $n=30$; obtain 192 or better |
|---|---|---|
## Part (iii):
| $20 \times 0.9^{p-1} < 0.4$, i.e. $0.9^{p-1} < 0.02$ | B1 | Correct $20 \times 0.9^{p-1}$ seen or implied |
|---|---|---|
| $(p-1)\log 0.9 < \log 0.02$ | M1 | Link to 0.4, rearrange to $0.9^k = c$, introduce logarithms, drop power, or equiv correct method |
| $p - 1 > \frac{\log 0.02}{\log 0.9}$ | M1 | Correct method for solving their (in)equation |
| $p > 38.1$, hence $p = 39$ | A1 **4** | State 39 (not inequality), no wrong working seen |
---6 A geometric progression has first term 20 and common ratio 0.9.\\
(i) Find the sum to infinity.\\
(ii) Find the sum of the first 30 terms.\\
(iii) Use logarithms to find the smallest value of $p$ such that the $p$ th term is less than 0.4 .
\hfill \mbox{\textit{OCR C2 2009 Q6 [8]}}