| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2007 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Shared terms between AP and GP |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard formulas for AP/GP terms and basic algebraic manipulation. Part (i) is pure recall, part (ii) involves setting up and solving a simple equation using the GP property, and part (iii) requires one additional calculation. The problem is slightly above average due to the connection between two sequences, but the steps are routine and well-signposted. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(a + 4d\) and \(a + 14d\) | B1 [1] | Both correct. |
| (ii) \(a+4d = ar\), \(a+14d=ar^2\) | M1 | Correct first step – award the mark for both of these starts. |
| or \(\frac{a}{a+4d} = \frac{a+4d}{a+14d}\) or "ac=b²" → \(3a = 8d\) | M1, A1 [3] | Correct elimination of \(r\) co. nb answer was given. |
| (iii) \(r = \frac{a+4d}{a}\) or \(\frac{a+14d}{a+4d} = 2.5\) | M1 A1 [2] | Statement + some substitution. co. |
**(i)** $a + 4d$ and $a + 14d$ | B1 [1] | Both correct.
**(ii)** $a+4d = ar$, $a+14d=ar^2$ | M1 | Correct first step – award the mark for both of these starts.
or $\frac{a}{a+4d} = \frac{a+4d}{a+14d}$ or "ac=b²" → $3a = 8d$ | M1, A1 [3] | Correct elimination of $r$ co. nb answer was given.
**(iii)** $r = \frac{a+4d}{a}$ or $\frac{a+14d}{a+4d} = 2.5$ | M1 A1 [2] | Statement + some substitution. co.4 The 1st term of an arithmetic progression is $a$ and the common difference is $d$, where $d \neq 0$.\\
(i) Write down expressions, in terms of $a$ and $d$, for the 5th term and the 15th term.
The 1st term, the 5th term and the 15th term of the arithmetic progression are the first three terms of a geometric progression.\\
(ii) Show that $3 a = 8 d$.\\
(iii) Find the common ratio of the geometric progression.
\hfill \mbox{\textit{CAIE P1 2007 Q4 [6]}}