| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| 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 direct application of standard formulas. Part (a) uses S∞ = a/(1-r) with given values, part (b) is trivial multiplication, and part (c) involves basic index manipulation of powers of 2 and 3. All three parts are routine recall with minimal problem-solving, making it easier than average but not trivial since it requires careful algebraic manipulation in part (c). |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(S_\infty = \dfrac{54}{1 - \frac{8}{9}}\) | M1 | Use of \(S_\infty = \dfrac{a}{1-r}\) |
| \(= \dfrac{54}{\frac{1}{9}} = 486\) | A1 |
| Answer | Marks |
|---|---|
| Second term \(= 54 \times \frac{8}{9} = 48\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(T_{12} = 54 \times \left(\dfrac{8}{9}\right)^{11}\) | M1 | Use of \(ar^{11}\) |
| \(= \dfrac{2 \times 3^3 \times 2^{33}}{3^{22}} = \dfrac{2^{34}}{3^{19}}\) | A1 | Correct manipulation to required form |
| So \(p = 34\), \(q = 19\) | A1 | Both correct integers stated |
# Question 3:
## Part (a):
| $S_\infty = \dfrac{54}{1 - \frac{8}{9}}$ | M1 | Use of $S_\infty = \dfrac{a}{1-r}$ |
| $= \dfrac{54}{\frac{1}{9}} = 486$ | A1 | |
## Part (b):
| Second term $= 54 \times \frac{8}{9} = 48$ | B1 | |
## Part (c):
| $T_{12} = 54 \times \left(\dfrac{8}{9}\right)^{11}$ | M1 | Use of $ar^{11}$ |
| $= \dfrac{2 \times 3^3 \times 2^{33}}{3^{22}} = \dfrac{2^{34}}{3^{19}}$ | A1 | Correct manipulation to required form |
| So $p = 34$, $q = 19$ | A1 | Both correct integers stated |3 The first term of a geometric series is 54 and the common ratio of the series is $\frac { 8 } { 9 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the sum to infinity of the series.
\item Find the second term of the series.
\item Show that the 12th term of the series can be written in the form $\frac { 2 ^ { p } } { 3 ^ { q } }$, where $p$ and $q$ are integers.\\[0pt]
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2014 Q3 [6]}}