| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2006 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find sum to infinity |
| Difficulty | Moderate -0.8 Part (a) is a straightforward arithmetic series problem requiring identification of first/last terms and application of the sum formula. Part (b)(i) involves simple geometric mean calculation (x² = 144×64), and (b)(ii) is direct application of the sum to infinity formula once r is found. All components are routine textbook exercises with no problem-solving insight required, making this easier than average but not trivial due to the multi-step nature. |
| 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 |
|---|---|---|
| (a) \(a = 105\) Either \(l = 399\) or \(d = 7\) \(n = 43\) \(\rightarrow 10836\) | B1 B1 B1 B1 [4] | co co co co |
| (b) \(r^2 = \frac{64}{144} \rightarrow r = \frac{3}{4}\) | M1 | award in either part |
| (i) Either \(x = \frac{ar}{1 - r} \rightarrow x = 96\) or \(\frac{144}{x} = \frac{x}{64} \rightarrow x = 96\) | M1 A1 | either method ok |
| (ii) Use of \(S_\infty = \frac{a}{1-r}\) | M1 | Used with his \(a\) and \(r\) |
| \(\rightarrow 432\) | A1 [5] | Co (nb do not penalise if \(r\) and / or \(x\) negative as well as positive.) |
(a) $a = 105$ Either $l = 399$ or $d = 7$ $n = 43$ $\rightarrow 10836$ | B1 B1 B1 B1 [4] | co co co co
(b) $r^2 = \frac{64}{144} \rightarrow r = \frac{3}{4}$ | M1 | award in either part
(i) Either $x = \frac{ar}{1 - r} \rightarrow x = 96$ or $\frac{144}{x} = \frac{x}{64} \rightarrow x = 96$ | M1 A1 | either method ok
(ii) Use of $S_\infty = \frac{a}{1-r}$ | M1 | Used with his $a$ and $r$
$\rightarrow 432$ | A1 [5] | Co (nb do not penalise if $r$ and / or $x$ negative as well as positive.)
---6
\begin{enumerate}[label=(\alph*)]
\item Find the sum of all the integers between 100 and 400 that are divisible by 7 .
\item The first three terms in a geometric progression are $144 , x$ and 64 respectively, where $x$ is positive. Find
\begin{enumerate}[label=(\roman*)]
\item the value of $x$,
\item the sum to infinity of the progression.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2006 Q6 [9]}}