| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Two related arithmetic progressions |
| Difficulty | Standard +0.3 Part (a) involves routine manipulation of GP formulas (sum to infinity and nth term) leading to a quadratic equation. Part (b) requires setting up two equations from given ratios of AP terms and sums, then solving simultaneous equations. Both parts are standard textbook exercises requiring formula recall and algebraic manipulation but no novel insight or complex problem-solving. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(ar = \frac{24}{100} \times \frac{a}{1-r}\) | M1 | Form an equation using a numerical form of the percentage and correct formula for \(u_2\) and \(S_\infty\) |
| \(100r^2 - 100r + 24 [=0]\) | A1 | OE. All 3 terms on one side of an equation |
| \((20r-8)(5r-3)[=0] \rightarrow r = \frac{2}{5}, \frac{3}{5}\) | A1 | Dependent on factors or formula seen from their quadratic |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3 \times \{(a+4d)\} = \{(2(a+1) + 11(d+1))\}\) | *M1 | SOI. Attempt to cross multiply with contents of at least one \(\{\}\) correct |
| Simplifies to \(a + d = 13\) | A1 | |
| \(\left[\frac{5}{2}\right] \times 3\{(2a+4d)\} = \left[\frac{5}{2}\right] \times 2\{(4(a+1) + 4(d+1))\}\) | *M1 | SOI. Attempt to cross multiply with contents of at least one \(\{\}\) correct |
| Simplifies to \(-a + 2d = 8\) | A1 | |
| Solve 2 linear equations simultaneously | DM1 | Elimination or substitution expected |
| \(d = 7,\ a = 6\) | A1 | SC B1 for \(a=6\), \(d=7\) without complete working |
## Question 9(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $ar = \frac{24}{100} \times \frac{a}{1-r}$ | M1 | Form an equation using a numerical form of the percentage and correct formula for $u_2$ and $S_\infty$ |
| $100r^2 - 100r + 24 [=0]$ | A1 | OE. All 3 terms on one side of an equation |
| $(20r-8)(5r-3)[=0] \rightarrow r = \frac{2}{5}, \frac{3}{5}$ | A1 | Dependent on factors or formula seen from their quadratic |
---
## Question 9(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3 \times \{(a+4d)\} = \{(2(a+1) + 11(d+1))\}$ | *M1 | SOI. Attempt to cross multiply with contents of at least one $\{\}$ correct |
| Simplifies to $a + d = 13$ | A1 | |
| $\left[\frac{5}{2}\right] \times 3\{(2a+4d)\} = \left[\frac{5}{2}\right] \times 2\{(4(a+1) + 4(d+1))\}$ | *M1 | SOI. Attempt to cross multiply with contents of at least one $\{\}$ correct |
| Simplifies to $-a + 2d = 8$ | A1 | |
| Solve 2 linear equations simultaneously | DM1 | Elimination or substitution expected |
| $d = 7,\ a = 6$ | A1 | **SC B1** for $a=6$, $d=7$ without complete working |
---9
\begin{enumerate}[label=(\alph*)]
\item A geometric progression is such that the second term is equal to $24 \%$ of the sum to infinity. Find the possible values of the common ratio.
\item An arithmetic progression $P$ has first term $a$ and common difference $d$. An arithmetic progression $Q$ has first term 2( $a + 1$ ) and common difference ( $d + 1$ ). It is given that
$$\frac { 5 \text { th term of } P } { 12 \text { th term of } Q } = \frac { 1 } { 3 } \quad \text { and } \quad \frac { \text { Sum of first } 5 \text { terms of } P } { \text { Sum of first } 5 \text { terms of } Q } = \frac { 2 } { 3 } .$$
Find the value of $a$ and the value of $d$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q9 [9]}}