| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 10 |
| 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 techniques: setting up equations from given conditions, solving simultaneous equations, and applying memorized formulas for GP sum to infinity and AP sum. The algebra is routine and the problem structure is typical of textbook exercises, making it slightly easier than average. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors |
(i) B1: $8 + 4d = 8r$ — co but allow if $a$ in place of 8
B1: $8 + 7d = 8r^2$ — co but allow if $a$ in place of 8
M1: Eliminates one of the variables — Complete elimination of either $r$ or $d$
DM1: — Correct method of solution
A1: $\to 4r^2 – 7r + 3 = 0$ Solution — nb answer for $r$ given
A1: $\to r = \frac{3}{4}$ $\to d = −\frac{1}{2}$ — (assumes $r = \frac{3}{4}$, give B1B1 for equations, B1 for $d$)
[6]
(ii) M1: $S_{\infty} = \frac{a}{1-r} \to 32$ — Correct formula used
A1: — Correct formula used
[2]
(iii) M1: $S_8 = 4(16 + 7d)$ — Correct formula used; $64 + 28d$ ok
A1: $= 50$
[3]8 The first term of an arithmetic progression is 8 and the common difference is $d$, where $d \neq 0$. The first term, the fifth term and the eighth term of this arithmetic progression are the first term, the second term and the third term, respectively, of a geometric progression whose common ratio is $r$.\\
(i) Write down two equations connecting $d$ and $r$. Hence show that $r = \frac { 3 } { 4 }$ and find the value of $d$.\\
(ii) Find the sum to infinity of the geometric progression.\\
(iii) Find the sum of the first 8 terms of the arithmetic progression.
\hfill \mbox{\textit{CAIE P1 2009 Q8 [10]}}