| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Relationship between two GPs |
| Difficulty | Standard +0.3 This question tests standard sum to infinity formula for GPs and sum of n terms for APs. Part (a) requires setting two GP sum formulas equal and solving algebraically—straightforward application with one constraint on r from convergence. Part (b) is similar but with AP formulas. Both parts are routine applications of memorized formulas with basic algebraic manipulation, slightly above average only due to the two-progression comparison setup. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{3a}{1-r} = \frac{a}{1+2r}\) | M1 | Attempt to equate 2 sums to infinity. At least one correct |
| \(3 + 6r = 1 - r\) | DM1 | Elimination of 1 variable (\(a\)) at any stage and multiplication |
| \(r = -\frac{2}{7}\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}n\big[2\times15+(n-1)4\big] = \frac{1}{2}n\big[2\times420+(n-1)(-5)\big]\) | M1A1 | Attempt to equate 2 sum to \(n\) terms, at least one correct (M1). Both correct (A1) |
| \(n = 91\) | A1 | |
| Total: 3 |
## Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{3a}{1-r} = \frac{a}{1+2r}$ | M1 | Attempt to equate 2 sums to infinity. At least one correct |
| $3 + 6r = 1 - r$ | DM1 | Elimination of 1 variable ($a$) at any stage and multiplication |
| $r = -\frac{2}{7}$ | A1 | |
| **Total: 3** | | |
---
## Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}n\big[2\times15+(n-1)4\big] = \frac{1}{2}n\big[2\times420+(n-1)(-5)\big]$ | M1A1 | Attempt to equate 2 sum to $n$ terms, at least one correct **(M1)**. Both correct **(A1)** |
| $n = 91$ | A1 | |
| **Total: 3** | | |
---3
\begin{enumerate}[label=(\alph*)]
\item A geometric progression has first term $3 a$ and common ratio $r$. A second geometric progression has first term $a$ and common ratio $- 2 r$. The two progressions have the same sum to infinity. Find the value of $r$.
\item The first two terms of an arithmetic progression are 15 and 19 respectively. The first two terms of a second arithmetic progression are 420 and 415 respectively. The two progressions have the same sum of the first $n$ terms. Find the value of $n$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2017 Q3 [6]}}