| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Mixed arithmetic and geometric |
| Difficulty | Standard +0.8 This question requires students to work with both geometric and arithmetic progressions in non-standard forms. Part (a) involves finding a common ratio from algebraic terms, applying sum to infinity conditions, and solving a resulting equation. Part (b) requires deriving the common difference from the given terms, setting up an inequality with the arithmetic series formula, and solving a quadratic inequality. The algebraic manipulation and multi-step reasoning elevate this above routine progression questions. |
| 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 |
| \(r = \dfrac{a}{a+2}\) | B1 | OE SOI |
| \(\dfrac{a}{1 - \dfrac{a}{a+2}} = 264\) | M1 | Use of \(S_\infty\) formula |
| \(\dfrac{a(a+2)}{a+2-a} = 264\) leading to \(\dfrac{a(a+2)}{2} = 264\) leading to \(a^2 + 2a - 528 = 0\) | M1* | Process to a 3 term quadratic or 3 term cubic. May contain terms on LHS and RHS |
| \((a-22)(a+24) = 0\) | DM1 | Attempt to solve |
| \(a = 22\) (only) | A1 | 22 without working SC DB1 (dep on 2nd M1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(d = \dfrac{6^2}{6+2} - 6 = -\dfrac{3}{2}\) | B1 | |
| \(\dfrac{n}{2}\left\{12 + (n-1)\left(\dfrac{-3}{2}\right)\right\} < -480\) | M1* | Forming an inequation with their numerical \(d\). May use an equality |
| \([3](n^2 - 9n - 640) > 0\) | A1 | OE May contain terms on LHS and RHS |
| \([n=] \dfrac{9 \pm \sqrt{81+2560}}{2}\) | DM1 | OE. Expect 30.19. Working for solution must be shown |
| 31 only | A1 | Must come from a correct first inequality (or equality). 31 no working SC DB1 (dep on correct quadratic and correct inequality/equality) |
## Question 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $r = \dfrac{a}{a+2}$ | B1 | OE SOI |
| $\dfrac{a}{1 - \dfrac{a}{a+2}} = 264$ | M1 | Use of $S_\infty$ formula |
| $\dfrac{a(a+2)}{a+2-a} = 264$ leading to $\dfrac{a(a+2)}{2} = 264$ leading to $a^2 + 2a - 528 = 0$ | M1* | Process to a 3 term quadratic or 3 term cubic. May contain terms on LHS and RHS |
| $(a-22)(a+24) = 0$ | DM1 | Attempt to solve |
| $a = 22$ (only) | A1 | 22 without working **SC DB1** (dep on 2nd M1) |
## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $d = \dfrac{6^2}{6+2} - 6 = -\dfrac{3}{2}$ | B1 | |
| $\dfrac{n}{2}\left\{12 + (n-1)\left(\dfrac{-3}{2}\right)\right\} < -480$ | M1* | Forming an inequation with their numerical $d$. May use an equality |
| $[3](n^2 - 9n - 640) > 0$ | A1 | OE May contain terms on LHS and RHS |
| $[n=] \dfrac{9 \pm \sqrt{81+2560}}{2}$ | DM1 | OE. Expect 30.19. Working for solution must be shown |
| 31 only | A1 | Must come from a correct first inequality (or equality). 31 no working **SC DB1** (dep on correct quadratic and correct inequality/equality) |8 A progression has first term $a$ and second term $\frac { a ^ { 2 } } { a + 2 }$, where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item For the case where the progression is geometric and the sum to infinity is 264 , find the value of $a$.
\item For the case where the progression is arithmetic and $a = 6$, determine the least value of $n$ required for the sum of the first $n$ terms to be less than - 480 .
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q8 [10]}}