| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Form and solve quadratic in parameter |
| Difficulty | Moderate -0.8 This is a straightforward two-part question testing basic arithmetic and geometric progression formulas. Part (a) requires direct application of AP formulas with given values. Part (b) involves forming a quadratic equation from the GP common ratio property (standard technique), then routine calculations for subsequent terms and sum to infinity. All steps are textbook exercises with no novel problem-solving 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 |
|---|---|---|
| Answer | Mark | Guidance |
| 21st term \(= 13 + 20 \times 1.2 = 37\) (km) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(S_{21} = \frac{1}{2} \times 21 \times (26 + 20 \times 1.2)\) or \(\frac{1}{2} \times 21 \times (13 + {\rm their}\ 37)\) | M1 | A correct sum formula used with correct values for \(a\), \(d\) and \(n\) |
| \(525\) (km) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{x-3}{x} = \frac{x-5}{x-3}\) oe (or use of \(a\), \(ar\) and \(ar^2\)) | M1 | Any valid method to obtain an equation in one variable |
| \((a =\) or \(x =)\ 9\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(r = \left(\frac{x-3}{x}\right)\) or \(\left(\frac{x-5}{x-3}\right)\) or \(\sqrt{\frac{x-5}{x}} = \frac{2}{3}\). Fourth term \(= 9 \times \left(\frac{2}{3}\right)^3\) | M1 | Any valid method to find \(r\) and the fourth term with their \(a\) & \(r\) |
| \(2\frac{2}{3}\) or \(2.67\) | A1 | OE, AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(S_\infty = \frac{a}{1-r} = \frac{9}{1-\frac{2}{3}}\) | M1 | Correct formula and using their \(r\) and \(a\), with \( |
| \(27\) or \(27.0\) | A1 | AWRT |
## Question 8(a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| 21st term $= 13 + 20 \times 1.2 = 37$ (km) | B1 | |
---
## Question 8(a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $S_{21} = \frac{1}{2} \times 21 \times (26 + 20 \times 1.2)$ or $\frac{1}{2} \times 21 \times (13 + {\rm their}\ 37)$ | M1 | A correct sum formula used with correct values for $a$, $d$ and $n$ |
| $525$ (km) | A1 | |
---
## Question 8(b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{x-3}{x} = \frac{x-5}{x-3}$ oe (or use of $a$, $ar$ and $ar^2$) | M1 | Any valid method to obtain an equation in one variable |
| $(a =$ or $x =)\ 9$ | A1 | |
---
## Question 8(b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $r = \left(\frac{x-3}{x}\right)$ or $\left(\frac{x-5}{x-3}\right)$ or $\sqrt{\frac{x-5}{x}} = \frac{2}{3}$. Fourth term $= 9 \times \left(\frac{2}{3}\right)^3$ | M1 | Any valid method to find $r$ and the fourth term with their $a$ & $r$ |
| $2\frac{2}{3}$ **or** $2.67$ | A1 | OE, AWRT |
---
## Question 8(b)(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $S_\infty = \frac{a}{1-r} = \frac{9}{1-\frac{2}{3}}$ | M1 | Correct formula and using their $r$ and $a$, with $|r| < 1$, to obtain a numerical answer |
| $27$ **or** $27.0$ | A1 | AWRT |8
\begin{enumerate}[label=(\alph*)]
\item Over a 21-day period an athlete prepares for a marathon by increasing the distance she runs each day by 1.2 km . On the first day she runs 13 km .
\begin{enumerate}[label=(\roman*)]
\item Find the distance she runs on the last day of the 21-day period.
\item Find the total distance she runs in the 21-day period.
\end{enumerate}\item The first, second and third terms of a geometric progression are $x , x - 3$ and $x - 5$ respectively.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $x$.
\item Find the fourth term of the progression.
\item Find the sum to infinity of the progression.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2019 Q8 [9]}}