| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| 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 geometric series question requiring only standard formula applications: finding r by division, applying S_∞ = a/(1-r), using S_n formula, and verifying the nth term formula. All parts are routine recall with no problem-solving or novel insight required, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(r = 16 \div 20 = 0.8\) | B1 (1 mark) | OE |
| (b) \(\frac{a}{1-r} = \frac{20}{1-0.8} = 100\) | M1, A1F (2 marks) | OE Using a correct formula with \(a = 20\) or \(r = c's 0.8\); ft on \(c's\) value of \(r\) provided \( |
| (c) \(\{S_{20}\} = \frac{a(1-r^{20})}{1-r} = 100(1-0.8^{20}) = 98.847[07\ldots]\) | M1, A1 (2 marks) | OE Using a correct formula with \(n = 20\); Condone > 3dp |
| (d) \(nth\) term \(= 20 \cdot r^{n-1} = 20(0.8)^{n-1} = 20 \times 0.8^n \times 0.8^{-1} = 25 \times 0.8^n\) | M1, A1 (2 marks) | Ft on \(c's\) \(r\). Award even if \(16^{-1}\) seen; CSO; AG |
**(a)** $r = 16 \div 20 = 0.8$ | B1 (1 mark) | OE
**(b)** $\frac{a}{1-r} = \frac{20}{1-0.8} = 100$ | M1, A1F (2 marks) | OE Using a correct formula with $a = 20$ or $r = c's 0.8$; ft on $c's$ value of $r$ provided $|r| < 1$
**(c)** $\{S_{20}\} = \frac{a(1-r^{20})}{1-r} = 100(1-0.8^{20}) = 98.847[07\ldots]$ | M1, A1 (2 marks) | OE Using a correct formula with $n = 20$; Condone > 3dp
**(d)** $nth$ term $= 20 \cdot r^{n-1} = 20(0.8)^{n-1} = 20 \times 0.8^n \times 0.8^{-1} = 25 \times 0.8^n$ | M1, A1 (2 marks) | Ft on $c's$ $r$. Award even if $16^{-1}$ seen; CSO; AG
**Total for Q3: 7 marks**
---3 A geometric series begins
$$20 + 16 + 12.8 + 10.24 + \ldots$$
\begin{enumerate}[label=(\alph*)]
\item Find the common ratio of the series.
\item Find the sum to infinity of the series.
\item Find the sum of the first 20 terms of the series, giving your answer to three decimal places.
\item Prove that the $n$th term of the series is $25 \times 0.8 ^ { n }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2008 Q3 [7]}}