| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Arithmetic progression with parameters |
| Difficulty | Standard +0.3 Part (i) requires setting up two sum to infinity formulas and solving a simple equation for r. Part (ii) involves standard arithmetic progression formulas with substitution. Both parts are routine applications of formulas with straightforward algebra, slightly above average due to the parameter manipulation in (i) and the 3n term in (ii), but still standard textbook exercises. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(S = \frac{a}{1-r},\quad 3S = \frac{a}{1-2r}\) | B1 | At least \(3S = \frac{a}{1-2r}\) |
| \(1 - r = 3 - 6r\) | M1 | Eliminate \(S\) |
| \(r = \frac{2}{5}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(7 + (n-1)d = 84\) and/or \(7 + (3n-1)d = 245\) | B1 | At least one of these equations seen |
| \([(n-1)d = 77,\ (3n-1)d = 238,\ 2nd = 161]\) | B1 | Two different seen — unsimplified ok |
| \(\frac{n-1}{3n-1} = \frac{77}{238}\) (must be from the correct \(u_n\) formula) | M1 | Or other attempt to elim \(d\). E.g. sub \(d = \frac{161}{2n}\) (if \(n\) is eliminated \(d\) must be found) |
| \(n = 23\quad (d = \frac{77}{22} = 3.5)\) | A1 |
## Question 7:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $S = \frac{a}{1-r},\quad 3S = \frac{a}{1-2r}$ | **B1** | At least $3S = \frac{a}{1-2r}$ |
| $1 - r = 3 - 6r$ | **M1** | Eliminate $S$ |
| $r = \frac{2}{5}$ | **A1** | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $7 + (n-1)d = 84$ and/or $7 + (3n-1)d = 245$ | **B1** | At least one of these equations seen |
| $[(n-1)d = 77,\ (3n-1)d = 238,\ 2nd = 161]$ | **B1** | Two different seen — unsimplified ok |
| $\frac{n-1}{3n-1} = \frac{77}{238}$ (must be from the correct $u_n$ formula) | **M1** | Or other attempt to elim $d$. E.g. sub $d = \frac{161}{2n}$ (if $n$ is eliminated $d$ must be found) |
| $n = 23\quad (d = \frac{77}{22} = 3.5)$ | **A1** | |
---7 (i) A geometric progression has first term $a ( a \neq 0 )$, common ratio $r$ and sum to infinity $S$. A second geometric progression has first term $a$, common ratio $2 r$ and sum to infinity $3 S$. Find the value of $r$.\\
(ii) An arithmetic progression has first term 7. The $n$th term is 84 and the ( $3 n$ )th term is 245 . Find the value of $n$.
\hfill \mbox{\textit{CAIE P1 2014 Q7 [7]}}