| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Relationship between two GPs |
| Difficulty | Standard +0.3 This question requires calculating sums to infinity using the standard formula S = a/(1-r), identifying an arithmetic progression pattern, and working backwards to find terms of a geometric sequence. While it involves multiple steps across two parts, each step uses routine A-level formulas with no novel insight required. The arithmetic progression connection is explicitly stated rather than needing to be discovered, making this slightly easier than a typical multi-part question. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(S_P = \frac{2}{1 - \frac{1}{2}}, S_P = \frac{3}{1 - \frac{1}{3}}\) | M1 | At least one correct |
| \(S_P = 4, S_Q = \frac{9}{2}\) | A1 | At least one correct |
| \(S_R = 5\) cao | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{4}{1-r} = \text{their } S_R\) | M1 | |
| \(r = \frac{1}{5}\) | A1 | [3] |
| \(R = 4 + \frac{4}{5} + \frac{4}{25} = 4\frac{24}{25}\) or 4.96 cao | A1 | [3] |
### (i)
$S_P = \frac{2}{1 - \frac{1}{2}}, S_P = \frac{3}{1 - \frac{1}{3}}$ | **M1** | At least one correct
$S_P = 4, S_Q = \frac{9}{2}$ | **A1** | At least one correct
$S_R = 5$ cao | **A1** | [3]
### (ii)
$\frac{4}{1-r} = \text{their } S_R$ | **M1** |
$r = \frac{1}{5}$ | **A1** | [3]
$R = 4 + \frac{4}{5} + \frac{4}{25} = 4\frac{24}{25}$ or 4.96 cao | **A1** | [3]
---Three geometric progressions, $P$, $Q$ and $R$, are such that their sums to infinity are the first three terms respectively of an arithmetic progression.
Progression $P$ is $2, 1, \frac{1}{2}, \frac{1}{4}, \ldots$
Progression $Q$ is $3, 1, \frac{1}{3}, \frac{1}{9}, \ldots$
\begin{enumerate}[label=(\roman*)]
\item Find the sum to infinity of progression $R$. [3]
\item Given that the first term of $R$ is 4, find the sum of the first three terms of $R$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2014 Q4 [6]}}