| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Relationship between two GPs |
| Difficulty | Standard +0.3 Part (a) is a routine AP problem requiring two simultaneous equations. Part (b) involves setting up sum to infinity formulas for two GPs and solving simultaneous equations, but the algebra is straightforward and the concept is standard. This is slightly above average difficulty due to the two-part structure and the need to handle two GPs simultaneously, but requires no novel insight. |
| 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/Working | Marks | Guidance |
| \(a + 5d = 23\) | B1 | Solution of 2 linear equations |
| \(5(2a + 9d) = 200\) | B1 | |
| Attempt solution, expect \(d = 6\), \(a = -7\) | M1 | |
| \(29\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{1-r} (=) \frac{4}{1 - \frac{1}{4}r}\) | M1 | Use of \(S_\infty\) formula twice |
| \(r = \frac{4}{5}\) oe \(S = 5\) | A1A1 | [3] |
## Question 6:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a + 5d = 23$ | B1 | Solution of 2 linear equations |
| $5(2a + 9d) = 200$ | B1 | |
| Attempt solution, expect $d = 6$, $a = -7$ | M1 | |
| $29$ | A1 | [4] |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{1-r} (=) \frac{4}{1 - \frac{1}{4}r}$ | M1 | Use of $S_\infty$ formula twice |
| $r = \frac{4}{5}$ oe $S = 5$ | A1A1 | [3] |
---6
\begin{enumerate}[label=(\alph*)]
\item The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200 . Find the seventh term.
\item A geometric progression has first term 1 and common ratio $r$. A second geometric progression has first term 4 and common ratio $\frac { 1 } { 4 } r$. The two progressions have the same sum to infinity, $S$. Find the values of $r$ and $S$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2011 Q6 [7]}}