| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 7 |
| 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 application of the sum to infinity formula after identifying r = 0.5² = 0.25 from the given terms. Part (b) requires finding the number of terms using the nth term formula, then applying the sum formula—standard multi-step arithmetic progression work. Both parts are routine textbook exercises requiring only direct formula application with no problem-solving insight. |
| 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) \(a = 0.5\), \(r = 0.5^2\) | B1 | For both \(a\) and \(r\) |
| Uses correct formula \(= 0.5 \div 0.75\) | M1 | Uses correct formula with some \(a\), \(r\) |
| \(\rightarrow S_{\infty} = \frac{2}{3}\) (or 0.667) | A1 [3] | co |
| (b) \(a = 5\), \(d = 4\) | ||
| Uses \(200 = a + (n-1)d\) or T.I. | M1 | Attempt at finding the number of terms |
| 50 terms in the progression | ||
| Use of correct Sum formula | M1 | |
| \(\rightarrow 5150\) | A1 [4] | Correct formula (could use the last term (201)) |
| co |
(a) $a = 0.5$, $r = 0.5^2$ | B1 | For both $a$ and $r$
Uses correct formula $= 0.5 \div 0.75$ | M1 | Uses correct formula with some $a$, $r$
$\rightarrow S_{\infty} = \frac{2}{3}$ (or 0.667) | A1 [3] | co
(b) $a = 5$, $d = 4$ | |
Uses $200 = a + (n-1)d$ or T.I. | M1 | Attempt at finding the number of terms
50 terms in the progression | |
Use of correct Sum formula | M1 |
$\rightarrow 5150$ | A1 [4] | Correct formula (could use the last term (201))
| | co7
\begin{enumerate}[label=(\alph*)]
\item Find the sum to infinity of the geometric progression with first three terms $0.5,0.5 ^ { 3 }$ and $0.5 ^ { 5 }$.
\item The first two terms in an arithmetic progression are 5 and 9. The last term in the progression is the only term which is greater than 200 . Find the sum of all the terms in the progression.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2009 Q7 [7]}}