| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Sum of first n terms |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard arithmetic progression formulas (sum, nth term) and basic geometric progression calculations. Part (a) requires direct application of S_n and a_n formulas with minimal problem-solving, while part (b) is routine compound interest/GP work. All techniques are textbook exercises with clear pathways and no novel insight required. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
**(a)**
- M1: $a = -15$, $n = 25$. Must be correct formula. co
- A1: Use of $S_n \Rightarrow d = 3$
**(i)**
- M1: Last term $= a + 24d$. Must be $a + 24d$
- A1: $\Rightarrow 57$ (or $525 = \frac{1}{2} \times 25 \times (-15 + l) \Rightarrow l = 57$). $\checkmark$ for his $d$
**(ii)**
- M1: Positive terms are $3, 6, \ldots, 57$. Either $a = 0$ or $3$, $n = 19$ or $20$
- A1: Use of $S_{19}$ or $S_{20}$ $\Rightarrow 570$. Correct use of formula for $S_n$
**(b)**
- B1: $r = 1.05$. co
- B1: In either part (i) or (ii)
**(i)**
- M1: $11$th term $= ar^{10} = \$6516$ or $\$6520$. co
- A1: co
**(ii)**
- M1: $S_{11} = \frac{4000 \times (1.05^{11} - 1)}{0.05}$. Correct sum formula with their $r$
- A1: $= \$56800$ or $(56827)$. co10
\begin{enumerate}[label=(\alph*)]
\item An arithmetic progression contains 25 terms and the first term is - 15 . The sum of all the terms in the progression is 525. Calculate
\begin{enumerate}[label=(\roman*)]
\item the common difference of the progression,
\item the last term in the progression,
\item the sum of all the positive terms in the progression.
\end{enumerate}\item A college agrees a sponsorship deal in which grants will be received each year for sports equipment. This grant will be $\$ 4000$ in 2012 and will increase by $5 \%$ each year. Calculate
\begin{enumerate}[label=(\roman*)]
\item the value of the grant in 2022,
\item the total amount the college will receive in the years 2012 to 2022 inclusive.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2011 Q10 [10]}}