| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Mixed arithmetic and geometric |
| Difficulty | Moderate -0.3 Part (i) is a straightforward application of the arithmetic series formula S_n = n/2[2a + (n-1)d] requiring simple algebraic manipulation. Part (ii) requires connecting AP and GP formulas and solving a quadratic equation, which is standard A-level technique but involves multiple steps. Overall, this is a routine multi-part question slightly easier than average due to the mechanical nature of the calculations with no novel insight required. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Uses \(S_n\); \(\frac{n}{2}(24 + 8d) = 135 \to d = \frac{3}{4}\) | M1, A1 | Uses correct formula; co |
| (ii) 9th term of AP \(= 12 + 8 \times \frac{3}{4} = 18\); GP 1st term 12, 2nd term 18; Common ratio \(= r = 18 \div 12 = 1\frac{1}{2}\); 3rd term of GP \(= ar^2 = 27\); nth term of AP is \(12 + (n-1)\frac{3}{4}\); \(12 + (n-1)\frac{3}{4} = 27 \to n = 21\) | B1, M1, M1, M1, A1 | ✓ on "d"; Uses "ar"; Uses \(ar^2\) or "ar" \(\times r\); Links AP with GP, co |
**(i)** Uses $S_n$; $\frac{n}{2}(24 + 8d) = 135 \to d = \frac{3}{4}$ | M1, A1 | Uses correct formula; co
**(ii)** 9th term of AP $= 12 + 8 \times \frac{3}{4} = 18$; GP 1st term 12, 2nd term 18; Common ratio $= r = 18 \div 12 = 1\frac{1}{2}$; 3rd term of GP $= ar^2 = 27$; nth term of AP is $12 + (n-1)\frac{3}{4}$; $12 + (n-1)\frac{3}{4} = 27 \to n = 21$ | B1, M1, M1, M1, A1 | ✓ on "d"; Uses "ar"; Uses $ar^2$ or "ar" $\times r$; Links AP with GP, coThe first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135.
\begin{enumerate}[label=(\roman*)]
\item Find the common difference of the progression. [2]
\end{enumerate}
The first term, the ninth term and the $n$th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the common ratio of the geometric progression and the value of $n$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q6 [7]}}