| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Shared terms between AP and GP |
| Difficulty | Standard +0.3 This is a straightforward multi-step problem requiring standard techniques: setting up equations from the GP condition (ratio equality), solving a simple quadratic, then applying standard sum formulas for AP and GP. While it involves multiple steps and two sequences, each step follows routine procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks | Guidance |
|---|---|---|
| 10(a) | State or imply that first 3 terms of GP are 5+d, 5+4d, 5+10d | B1 |
| Form equation (5+4d)2 =(5+d)(5+10d) or equivalent | M1 | |
| Obtain d =2.5 | A1 | Ignore 0 as a solution. |
| Answer | Marks |
|---|---|
| State or imply that first 3 terms of GP are 5+d, 5+4d, 5+10d | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| R−4 | M1 | OE |
| Answer | Marks |
|---|---|
| Obtain d =2.5 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 10(b) | Use correct formula for sum of AP with their value of d | M1 |
| Obtain or imply 7700 | A1 | |
| State or imply GP is 7.5, 15, 30,... | B1 | |
| Use correct formula for sum of GP with their common ratio | M1 |
| Answer | Marks |
|---|---|
| 77 10 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 10:
--- 10(a) ---
10(a) | State or imply that first 3 terms of GP are 5+d, 5+4d, 5+10d | B1
Form equation (5+4d)2 =(5+d)(5+10d) or equivalent | M1
Obtain d =2.5 | A1 | Ignore 0 as a solution.
SC B1 Obtain d =2.5 and 7.5, 15, 30 by trial and
improvement www.
Alternative Method for Question 10(a):
State or imply that first 3 terms of GP are 5+d, 5+4d, 5+10d | B1
5−5R
(5 + d)R = 5 + 4d → d = , (5+d)R2 =5+10 d → R2 – 3R + 2 [= 0]
R−4 | M1 | OE
Eliminates d.
Obtain d =2.5 | A1
3
Question | Answer | Marks | Guidance
--- 10(b) ---
10(b) | Use correct formula for sum of AP with their value of d | M1
Obtain or imply 7700 | A1
State or imply GP is 7.5, 15, 30,... | B1
Use correct formula for sum of GP with their common ratio | M1
Obtain S −G =27.5
77 10 | A1
5
Question | Answer | Marks | GuidanceAn arithmetic progression has first term 5 and common difference $d$, where $d > 0$. The second, fifth and eleventh terms of the arithmetic progression, in that order, are the first three terms of a geometric progression.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $d$. [3]
\item The sum of the first 77 terms of the arithmetic progression is denoted by $S_{77}$. The sum of the first 10 terms of the geometric progression is denoted by $G_{10}$.
Find the value of $S_{77} - G_{10}$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q10 [8]}}