| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Convergence conditions |
| Difficulty | Standard +0.3 This is a straightforward geometric series question requiring recall of the sum to infinity formula and solving a simple inequality. Part (i) is algebraic manipulation with the formula S = a/(1-r), and part (ii) requires recognizing that |r| < 1 for convergence, then finding the range of 2-r. Both parts are routine applications with no novel insight needed, making this slightly easier than average. |
| Spec | 1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks |
|---|---|
| 2(i) | r2 −3r+2 |
| Answer | Marks |
|---|---|
| 1−r | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 1−r | A1 | AG Factors must be shown. Expressions requiring minus sign taken out must |
| Answer | Marks |
|---|---|
| Total: | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| 2(ii) | Single range 1<S <3 or (1, 3) | B2 |
| Answer | Marks | Guidance |
|---|---|---|
| Total: | 2 | |
| Question | Answer | Marks |
Question 2:
--- 2(i) ---
2(i) | r2 −3r+2
S =
1−r | M1
( )( ) ( )( )
r−1 r−2 − 1−r r−2
S = = =2−rOR
1−r 1−r
( )( )
1−r 2−r
=2−r OE
1−r | A1 | AG Factors must be shown. Expressions requiring minus sign taken out must
be shown
Total: | 2
--- 2(ii) ---
2(ii) | Single range 1<S <3 or (1, 3) | B2 | Accept 1<2−r<3.
Correct range but with S = 2 omitted scores SR B1
1(cid:45)S(cid:45)3 scores SR B1.
[S > 1 and S < 3] scores SR B1.
Total: | 2
Question | Answer | Marks | GuidanceThe common ratio of a geometric progression is $r$. The first term of the progression is $(r^2 - 3r + 2)$ and the sum to infinity is $S$.
\begin{enumerate}[label=(\roman*)]
\item Show that $S = 2 - r$. [2]
\item Find the set of possible values that $S$ can take. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2017 Q2 [4]}}