| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | New GP from transformation |
| Difficulty | Moderate -0.8 Part (a) is a straightforward arithmetic progression problem requiring only the formula for nth term and sum. Part (b) involves standard geometric series sum to infinity formulas, with the slight twist of recognizing that the even-numbered terms form another GP with first term 12 and ratio 1/4. Both parts are routine applications of memorized formulas with minimal problem-solving required, making this easier than average. |
| 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 | Marks | Guidance |
| Substitute \(n=10\) and \(a=6\) into \(u_n = a+(n-1)d\) | \*M1 | Expect \(6+9d=19.5\) or equivalent. |
| \([d=]1.5\) | A1 | |
| Substitute \(a=6\) and *their* \(d\) into correct formula for the sum of 100 terms | DM1 | |
| Obtain \(8025\) | A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain \(S=48\) | B1 | |
| Identify for \(S_E\) first term 12 and common ratio \(\frac{1}{4}\) | B1 | |
| Attempt sum to infinity, \(S_E\), with at least one of first term and common ratio correct | M1 | Only awarded if \( |
| Obtain \(S_E = 16\) | A1 | |
| Total: 4 |
## Question 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $n=10$ and $a=6$ into $u_n = a+(n-1)d$ | \*M1 | Expect $6+9d=19.5$ or equivalent. |
| $[d=]1.5$ | A1 | |
| Substitute $a=6$ and *their* $d$ into correct formula for the sum of 100 terms | DM1 | |
| Obtain $8025$ | A1 | |
| **Total: 4** | | |
---
## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain $S=48$ | B1 | |
| Identify for $S_E$ first term 12 and common ratio $\frac{1}{4}$ | B1 | |
| Attempt sum to infinity, $S_E$, with at least one of first term and common ratio correct | M1 | Only awarded if $|r|<1$ |
| Obtain $S_E = 16$ | A1 | |
| **Total: 4** | | |
---8
\begin{enumerate}[label=(\alph*)]
\item An arithmetic progression is such that its first term is 6 and its tenth term is 19.5 .\\
Find the sum of the first 100 terms of this arithmetic progression.
\item A geometric progression $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is such that $a _ { 1 } = 24$ and the common ratio is $\frac { 1 } { 2 }$.
The sum to infinity of this geometric progression is denoted by $S$. The sum to infinity of the even-numbered terms (i.e. $a _ { 2 } , a _ { 4 } , a _ { 6 } , \ldots$ ) is denoted by $S _ { E }$.
Find the values of $S$ and $S _ { E }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q8 [8]}}