| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Recursive sequence definition |
| Difficulty | Easy -1.2 This is a straightforward geometric progression question requiring only direct application of the recursive formula to find terms, then using the standard sum formula. Both parts involve routine calculations with no problem-solving or conceptual challenges—simpler than average A-level questions. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(u_2 = 12\); \(u_3 = 9.6\), \(u_4 = 7.68\) (or any exact equivalents) | B1, B1∨ 2 | State \(u_2 = 12\). Correct \(u_3\) and \(u_4\) from their \(u_2\) |
| (ii) \(S_{20} = \frac{15(1-0.8^{20})}{1-0.8} = 74.1\) | M1, A1, A1 3 | Attempt use of \(S_n = \frac{a(1-r^n)}{1-r}\) with \(n = 20\) or \(19\). Obtain correct unsimplified expression. Obtain 74.1 or better. |
| Answer | Marks |
|---|---|
| M1, A2 | Last all 20 terms of GP. Obtain 74.1 |
(i) $u_2 = 12$; $u_3 = 9.6$, $u_4 = 7.68$ (or any exact equivalents) | B1, B1∨ 2 | State $u_2 = 12$. Correct $u_3$ and $u_4$ from their $u_2$
(ii) $S_{20} = \frac{15(1-0.8^{20})}{1-0.8} = 74.1$ | M1, A1, A1 3 | Attempt use of $S_n = \frac{a(1-r^n)}{1-r}$ with $n = 20$ or $19$. Obtain correct unsimplified expression. Obtain 74.1 or better.
**OR**
| M1, A2 | Last all 20 terms of GP. Obtain 74.11 A geometric progression $\mathrm { u } _ { 1 } , \mathrm { u } _ { 2 } , \mathrm { u } _ { 3 } , \ldots$ is defined by
$$u _ { 1 } = 15 \quad \text { and } \quad u _ { n + 1 } = 0.8 u _ { n } \text { for } n \geqslant 1 .$$
(i) Write down the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$.\\
(ii) Find $\sum _ { n = 1 } ^ { 20 } u _ { n }$.
\hfill \mbox{\textit{OCR C2 2007 Q1 [5]}}