| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| 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 GP (finding r from a₁ and a₄, sum to infinity) and AP (finding common difference and S₁₀). The connection between the two sequences is explicitly stated, requiring only substitution rather than insight. Slightly above average difficulty due to the multi-step nature and linking two sequences, but all techniques are routine. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(a = 81, ar^3 = 24\) | M1 | Valid method for \(r\). |
| \(\rightarrow r^3 = \frac{24}{81} \rightarrow r = \frac{2}{3}\) or 0.667 | A1 | co [2] |
| (ii) \(S_{\infty} = \frac{a}{1 - r} = 81 \div \frac{1}{3} = 243\) | M1 A1√ | Correct formula. √ for his \(a\) and \(r\), providing \(-1 < r < 1\). [2] |
| (iii) 2nd term of GP \(= ar = 81 \times \frac{2}{3} = 54\) | M1 | Finding the 2nd and 3rd terms of GP. |
| 3rd term of GP \(= ar^2 = 36\) | M1 | M for finding \(d\) + correct \(S_{10}\) formula. co |
| \(\rightarrow 3d = -18\) \((d = -6)\) | A1 | [3] |
| \(\rightarrow S_{10} = 5 \times (108 - 54) = 270\) |
**(i)** $a = 81, ar^3 = 24$ | M1 | Valid method for $r$.
$\rightarrow r^3 = \frac{24}{81} \rightarrow r = \frac{2}{3}$ or 0.667 | A1 | co [2]
**(ii)** $S_{\infty} = \frac{a}{1 - r} = 81 \div \frac{1}{3} = 243$ | M1 A1√ | Correct formula. √ for his $a$ and $r$, providing $-1 < r < 1$. [2]
**(iii)** 2nd term of GP $= ar = 81 \times \frac{2}{3} = 54$ | M1 | Finding the 2nd and 3rd terms of GP.
3rd term of GP $= ar^2 = 36$ | M1 | M for finding $d$ + correct $S_{10}$ formula. co
$\rightarrow 3d = -18$ $(d = -6)$ | A1 | [3]
$\rightarrow S_{10} = 5 \times (108 - 54) = 270$ |7 The first term of a geometric progression is 81 and the fourth term is 24 . Find\\
(i) the common ratio of the progression,\\
(ii) the sum to infinity of the progression.
The second and third terms of this geometric progression are the first and fourth terms respectively of an arithmetic progression.\\
(iii) Find the sum of the first ten terms of the arithmetic progression.
\hfill \mbox{\textit{CAIE P1 2008 Q7 [7]}}