| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Shared terms between AP and GP |
| Difficulty | Standard +0.8 Part (a) is routine application of the formula relating S_n to first term and common difference. Part (b) requires finding the GP common ratio, then setting up and solving an equation linking AP and GP terms through their position formulas—this involves algebraic manipulation across two sequences and is more conceptually demanding than standard single-sequence problems. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(S_1 = 10 = a\) | B1 | |
| \(S_2 = 24 = a + (a+d)\), \(d = 4\) | M1 A1 [3] | Correct use of \(S_n\) formula |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| GP: \(a = 64\), \(ar = 48 \rightarrow r = \frac{3}{4}\) | B1 | |
| \(\rightarrow\) 3rd term is \(ar^2 = 36\) | M1 | \(ar^2\) numerical – for their \(r\) |
| AP: \(a = 64\), \(a + 8d = 48 \rightarrow d = -2\) | B1 | |
| \(36 = 64 + (n-1)(-2)\) | M1 | Correct use of \(a+(n-1)d\) |
| \(\rightarrow n = 15\) | A1 [5] |
## Question 9:
$S_n = 2n^2 + 8n$
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $S_1 = 10 = a$ | B1 | |
| $S_2 = 24 = a + (a+d)$, $d = 4$ | M1 A1 [3] | Correct use of $S_n$ formula |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| GP: $a = 64$, $ar = 48 \rightarrow r = \frac{3}{4}$ | B1 | |
| $\rightarrow$ 3rd term is $ar^2 = 36$ | M1 | $ar^2$ numerical – for their $r$ |
| AP: $a = 64$, $a + 8d = 48 \rightarrow d = -2$ | B1 | |
| $36 = 64 + (n-1)(-2)$ | M1 | Correct use of $a+(n-1)d$ |
| $\rightarrow n = 15$ | A1 [5] | |
---9
\begin{enumerate}[label=(\alph*)]
\item In an arithmetic progression, the sum, $S _ { n }$, of the first $n$ terms is given by $S _ { n } = 2 n ^ { 2 } + 8 n$. Find the first term and the common difference of the progression.
\item The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the 1st term, the 9th term and the $n$th term respectively of an arithmetic progression. Find the value of $n$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2013 Q9 [8]}}