| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Compound growth applications |
| Difficulty | Moderate -0.3 This is a straightforward application of geometric sequences requiring identification of the common ratio (r=0.8), using the nth term formula to verify n=9, calculating S_20, and recognizing S_∞. All steps are routine with no conceptual challenges beyond standard GP formulas, making it slightly easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(r=0.8\) | B1 | SOI |
| \(50\times(\text{their }0.8)^7=10.5\) | M1 | Evaluate \(8^{\text{th}}\) or \(9^{\text{th}}\) term in GP. |
| \(50\times(\text{their }0.8)^8=8.39\). Hence 9th impact required. | A1 | AG Two terms correct + conclusion (mention of \(9^{\text{th}}\) impact or \(u_9\) somewhere in the solution). Statement that one is \(<10\) (and the other \(>10\)) is insufficient unless it mentions \(9^{\text{th}}\) impact or \(u_9\). |
| Alternative: Logarithm method | ||
| \(50\times(\text{their }0.8)^n<10 \Rightarrow (\text{their }0.8)^n<0.5\) | M1 | |
| \(n\log(\text{their }0.8)<\log0.5\) | ||
| \(n>\frac{\log0.5}{\log(\text{their }0.8)}\Rightarrow[n>]7.2\) | ||
| \(n=8\) hence \(9^{\text{th}}\) impact required | A1 | AG Need conclusion that mentions \(9^{\text{th}}\) impact or \(u_9\). |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{50(1-(\text{their }0.8)^{20})}{1-\text{their }0.8}\) | M1 | OE Use of formula with *their* \(r\) SOI. |
| \(=247\) | A1 | Must be to the nearest mm (not 247.1). |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{50}{1-\text{their }0.8}\) | M1 | Use of sum to infinity formula with *their* \(r\) SOI. Substituting a value of \(n\) into the sum formula M0. |
| \(=250\) | A1 | |
| 2 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $r=0.8$ | **B1** | SOI |
| $50\times(\text{their }0.8)^7=10.5$ | **M1** | Evaluate $8^{\text{th}}$ or $9^{\text{th}}$ term in GP. |
| $50\times(\text{their }0.8)^8=8.39$. Hence 9th impact required. | **A1** | **AG** Two terms correct + conclusion (mention of $9^{\text{th}}$ impact or $u_9$ somewhere in the solution). Statement that one is $<10$ (and the other $>10$) is insufficient unless it mentions $9^{\text{th}}$ impact or $u_9$. |
| **Alternative: Logarithm method** | | |
| $50\times(\text{their }0.8)^n<10 \Rightarrow (\text{their }0.8)^n<0.5$ | **M1** | |
| $n\log(\text{their }0.8)<\log0.5$ | | |
| $n>\frac{\log0.5}{\log(\text{their }0.8)}\Rightarrow[n>]7.2$ | | |
| $n=8$ hence $9^{\text{th}}$ impact required | **A1** | **AG** Need conclusion that mentions $9^{\text{th}}$ impact or $u_9$. |
| | **3** | |
---
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{50(1-(\text{their }0.8)^{20})}{1-\text{their }0.8}$ | **M1** | OE Use of formula with *their* $r$ SOI. |
| $=247$ | **A1** | Must be to the nearest mm (not 247.1). |
| | **2** | |
---
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{50}{1-\text{their }0.8}$ | **M1** | Use of sum to infinity formula with *their* $r$ SOI. Substituting a value of $n$ into the sum formula M0. |
| $=250$ | **A1** | |
| | **2** | |7 A tool for putting fence posts into the ground is called a 'post-rammer'. The distances in millimetres that the post sinks into the ground on each impact of the post-rammer follow a geometric progression. The first three impacts cause the post to sink into the ground by $50 \mathrm {~mm} , 40 \mathrm {~mm}$ and 32 mm respectively.
\begin{enumerate}[label=(\alph*)]
\item Verify that the 9th impact is the first in which the post sinks less than 10 mm into the ground.
\item Find, to the nearest millimetre, the total depth of the post in the ground after 20 impacts.
\item Find the greatest total depth in the ground which could theoretically be achieved.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q7 [7]}}