| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Sum of multiples or integers |
| Difficulty | Moderate -0.8 This question tests standard arithmetic and geometric series formulas with straightforward application. Part (a) requires identifying an arithmetic sequence and applying the sum formula—routine bookwork. Part (b) involves solving a simple equation for the first term and applying the sum to infinity formula. All steps are direct applications of memorized formulas with no problem-solving insight required, making it easier than average. |
| 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) \(a = 100\), \(d = 5\), \(n = 41 \rightarrow S = 8200\) | B1, M1 A1 [3] | co. Use of correct sum formula. co. |
| (b) (i) \(a + ar + ar^2\) or \(a\frac{(1-r^3)}{1-r} = 35 \rightarrow a = 45\) | B1, M1 A1 [3] | co. Solution of equation. co. |
| (ii) \(S_∞ = \frac{a}{1-r} = 27\) | M1 A1√ [2] | Correct use of formula. √ for his \(a\). |
(a) $a = 100$, $d = 5$, $n = 41 \rightarrow S = 8200$ | B1, M1 A1 [3] | co. Use of correct sum formula. co.
(b) (i) $a + ar + ar^2$ or $a\frac{(1-r^3)}{1-r} = 35 \rightarrow a = 45$ | B1, M1 A1 [3] | co. Solution of equation. co.
(ii) $S_∞ = \frac{a}{1-r} = 27$ | M1 A1√ [2] | Correct use of formula. √ for his $a$.\begin{enumerate}[label=(\alph*)]
\item Find the sum of all the multiples of 5 between 100 and 300 inclusive. [3]
\item A geometric progression has a common ratio of $-\frac{2}{3}$ and the sum of the first 3 terms is 35. Find
\begin{enumerate}[label=(\roman*)]
\item the first term of the progression, [3]
\item the sum to infinity. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2010 Q7 [8]}}