| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Real-world AP: find term or total |
| Difficulty | Moderate -0.8 This question tests standard arithmetic sequence formulas (finding nth term and sum) and basic geometric progression (finding first term and sum to infinity). Part (a) requires straightforward application of a_n = a_1 + (n-1)d with simple arithmetic, while part (b) involves routine GP formulas. Both parts are textbook exercises requiring only direct formula application with no problem-solving insight or complex manipulation. |
| 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) (i) \(a = 300\), \(d = 12\) \(\to 540 = 300 + (n-1)12 \to n = 21\) | M1 A1 [2] | Use of nth term. Ans 20 gets 0. Ignore incorrect units. Correct use of \(s_n\) formula. |
| (ii) \(S_{26} = 13(600 + 25 \times 12) = 11700 \to 3\) hours 15 minutes. | M1, A1 [2] | Correct use of \(s_n\) formula. |
| (b) \(ar = 48\) and \(ar^2 = 32 \to r = \frac{3}{4} \to a = 72\). \(S_\infty = 72 \div \frac{1}{3} = 216\). | M1, A1, M1, A1✓ [4] | Needs \(ar\) and \(ar^2\) + attempt at \(a\) and \(r\). Correct \(S_\infty\) formula with \( |
(a) (i) $a = 300$, $d = 12$ $\to 540 = 300 + (n-1)12 \to n = 21$ | M1 A1 [2] | Use of nth term. Ans 20 gets 0. Ignore incorrect units. Correct use of $s_n$ formula.
(ii) $S_{26} = 13(600 + 25 \times 12) = 11700 \to 3$ hours 15 minutes. | M1, A1 [2] | Correct use of $s_n$ formula.
(b) $ar = 48$ and $ar^2 = 32 \to r = \frac{3}{4} \to a = 72$. $S_\infty = 72 \div \frac{1}{3} = 216$. | M1, A1, M1, A1✓ [4] | Needs $ar$ and $ar^2$ + attempt at $a$ and $r$. Correct $S_\infty$ formula with $|r| < 1$.7
\begin{enumerate}[label=(\alph*)]
\item An athlete runs the first mile of a marathon in 5 minutes. His speed reduces in such a way that each mile takes 12 seconds longer than the preceding mile.
\begin{enumerate}[label=(\roman*)]
\item Given that the $n$th mile takes 9 minutes, find the value of $n$.
\item Assuming that the length of the marathon is 26 miles, find the total time, in hours and minutes, to complete the marathon.
\end{enumerate}\item The second and third terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2013 Q7 [8]}}