| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Sequence defined by formula |
| Difficulty | Moderate -0.8 Part (a) requires routine application of the formula relating S_n to first term and common difference (finding a and d from S_1 and S_2, or using S_n = n/2[2a + (n-1)d]). Part (b) involves setting up two simultaneous equations from the given conditions about a GP and solving them, which is standard but requires more algebraic manipulation. Both parts are straightforward applications of well-practiced techniques with no novel problem-solving required, making this easier than average. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(S_n = n^2 + 8n\) | ||
| \(S_1 = 9 \to a = 9\) | B1 M1 A1 | co Realises that \(S_2 = a + (a + d)\). co [3] |
| \(S_2 = 20 \to a + d = 11 \to d = 2\) | ||
| (or equating \(n^2 + 8n\) with \(S_n\) and comparing coefficients) | ||
| (b) \(a - ar = 9\) | B1 | co |
| \(ar + ar^2 = 30\) | B1 | co |
| Eliminates \(a \to 3r^2 + 13r - 10 = 0\) | M1 | Complete elimination of r or a. Correct quadratic. |
| or \(\to 2a^2 - 57a + 81 = 0\) | ||
| \(\to r = \frac{2}{3}\) | A1 | |
| \(\to a = 27\) | A1 | co (condone 27 or 1.5) [5] |
(a) $S_n = n^2 + 8n$ | |
$S_1 = 9 \to a = 9$ | B1 M1 A1 | co Realises that $S_2 = a + (a + d)$. co [3]
$S_2 = 20 \to a + d = 11 \to d = 2$ | |
(or equating $n^2 + 8n$ with $S_n$ and comparing coefficients) | |
(b) $a - ar = 9$ | B1 | co
$ar + ar^2 = 30$ | B1 | co
Eliminates $a \to 3r^2 + 13r - 10 = 0$ | M1 | Complete elimination of r or a. Correct quadratic.
or $\to 2a^2 - 57a + 81 = 0$ | |
$\to r = \frac{2}{3}$ | A1 |
$\to a = 27$ | A1 | co (condone 27 or 1.5) [5]\begin{enumerate}[label=(\alph*)]
\item In an arithmetic progression, the sum of the first $n$ terms, denoted by $S_n$, is given by
$$S_n = n^2 + 8n.$$
Find the first term and the common difference. [3]
\item In a geometric progression, the second term is $9$ less than the first term. The sum of the second and third terms is $30$. Given that all the terms of the progression are positive, find the first term. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q7 [8]}}