| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2008 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find common ratio from terms |
| Difficulty | Moderate -0.3 This is a straightforward geometric series question requiring standard formula application: using ar^n = term to set up two equations, solving for r, then finding a, and finally applying the sum formula. While it involves multiple steps (6 marks total), each step follows directly from memorized formulas with no conceptual challenges or problem-solving insight required. Slightly easier than average due to its routine nature. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks |
|---|---|
| (a) Complete method, using terms of form \(ar^n\), to find \(r\) | M1 |
| [e.g. Dividing \(ar^6 = 80\) by \(ar^3 = 10\) to find \(r\); \(r^6 - r^3 = 8\) is M0] | |
| \(r = 2\) | A1 (2) |
| Answer | Marks |
|---|---|
| (b) Complete method for finding \(a\) | M1 |
| [e.g. Substituting value for \(r\) into equation of form \(ar^k = 10\) or \(80\) and finding a value for \(a\)] | |
| \((8a = 10)\) \(a = \frac{5}{4} = 1\frac{1}{4}\) (equivalent single fraction or 1.25) | A1 (2) |
| (c) Substituting their values of \(a\) and \(r\) into correct formula for sum. | M1 |
| \[S = \frac{a(r^n-1)}{r-1} = \frac{5}{4}(2^{20}-1)\] (= 1310718.75) \(1\,310\,719\) (only this) | A1 (2) [6] |
**(a)** Complete method, using terms of form $ar^n$, to find $r$ | M1 |
[e.g. Dividing $ar^6 = 80$ by $ar^3 = 10$ to find $r$; $r^6 - r^3 = 8$ is M0] | |
$r = 2$ | A1 (2) |
**Notes:** M1: Condone errors in powers, e.g. $ar^4 = 10$ and/or $ar^7 = 80$
**(b)** Complete method for finding $a$ | M1 |
[e.g. Substituting value for $r$ into equation of form $ar^k = 10$ or $80$ and finding a value for $a$] | |
$(8a = 10)$ $a = \frac{5}{4} = 1\frac{1}{4}$ (equivalent single fraction or 1.25) | A1 (2) |
**(c)** Substituting their values of $a$ and $r$ into **correct formula for sum**. | M1 |
$$S = \frac{a(r^n-1)}{r-1} = \frac{5}{4}(2^{20}-1)$$ (= 1310718.75) $1\,310\,719$ (only this) | A1 (2) [6] |
**Notes:**
- A1: For $r = 2$, allow even if $ar^4 = 10$ and $ar^7 = 80$ used (just these)
- M1: Allow for numerical approach: e.g. $\frac{10}{r_c^3} \leftarrow \frac{10}{r_c^2} \leftarrow \frac{10}{r_c} \leftarrow 10$
- (c) Attempt 20 terms of series and add is M1 (correct last term 655360)
- If formula not quoted, errors in applying their $a$ and/or $r$ is M0
- Allow full marks for correct answer with no working seen.
---The fourth term of a geometric series is 10 and the seventh term of the series is 80.
For this series, find
\begin{enumerate}[label=(\alph*)]
\item the common ratio, [2]
\item the first term, [2]
\item the sum of the first 20 terms, giving your answer to the nearest whole number. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2008 Q2 [6]}}