| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find sum to infinity |
| Difficulty | Moderate -0.3 Part (a) is a straightforward application of geometric progression formulas: finding r from a₁ and a₃, then using the sum to infinity formula. Part (b)(i) requires setting up equations from the AP common difference property and solving a trigonometric equation, while (b)(ii) is direct substitution into the arithmetic series formula. All techniques are standard for P1 level with no novel insights required, making this slightly 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|<11.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = 50\), \(ar^2 = 32\) | B1 | Seen or implied |
| \(\rightarrow r = \frac{4}{5}\) (allow \(-\frac{4}{5}\) for M mark) | M1 | Finding \(r\) and use of correct \(S_\infty\) formula |
| \(\rightarrow S_\infty = 250\) | A1 | Only if \( |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2\sin x\), \(3\cos x\), \((\sin x + 2\cos x)\); \(3c - 2s = (s + 2c) - 3c\) (or uses \(a\), \(a+d\), \(a+2d\)) | M1 | Links terms up with AP, needs one expression for \(d\) |
| \(\rightarrow 4c = 3s \rightarrow t = \frac{4}{3}\) | M1 A1 | Arrives at \(t = k\) ag |
| SC uses \(t = \frac{4}{3}\) to show \(u_1 = \frac{8}{5}, u_2 = \frac{9}{5}, u_3 = \frac{10}{5}\) | B1 only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(c = \frac{3}{5}\), \(s = \frac{4}{5}\) or calculator \(x = 53.1°\) | M1 | |
| \(\rightarrow a = 1.6\), \(d = 0.2\) | M1 | Correct method for both \(a\) and \(d\) |
| \(\rightarrow S_{20} = 70\) | A1 | Uses \(S_n\) formula |
## Question 9(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = 50$, $ar^2 = 32$ | **B1** | Seen or implied |
| $\rightarrow r = \frac{4}{5}$ (allow $-\frac{4}{5}$ for M mark) | **M1** | Finding $r$ and use of correct $S_\infty$ formula |
| $\rightarrow S_\infty = 250$ | **A1** | Only if $|r| < 1$ |
## Question 9(b)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2\sin x$, $3\cos x$, $(\sin x + 2\cos x)$; $3c - 2s = (s + 2c) - 3c$ (or uses $a$, $a+d$, $a+2d$) | **M1** | Links terms up with AP, needs one expression for $d$ |
| $\rightarrow 4c = 3s \rightarrow t = \frac{4}{3}$ | **M1 A1** | Arrives at $t = k$ ag |
| SC uses $t = \frac{4}{3}$ to show $u_1 = \frac{8}{5}, u_2 = \frac{9}{5}, u_3 = \frac{10}{5}$ | **B1 only** | |
## Question 9(b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $c = \frac{3}{5}$, $s = \frac{4}{5}$ or calculator $x = 53.1°$ | **M1** | |
| $\rightarrow a = 1.6$, $d = 0.2$ | **M1** | Correct method for both $a$ and $d$ |
| $\rightarrow S_{20} = 70$ | **A1** | Uses $S_n$ formula |9
\begin{enumerate}[label=(\alph*)]
\item The first term of a geometric progression in which all the terms are positive is 50 . The third term is 32 . Find the sum to infinity of the progression.
\item The first three terms of an arithmetic progression are $2 \sin x , 3 \cos x$ and ( $\sin x + 2 \cos x$ ) respectively, where $x$ is an acute angle.
\begin{enumerate}[label=(\roman*)]
\item Show that $\tan x = \frac { 4 } { 3 }$.
\item Find the sum of the first twenty terms of the progression.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2016 Q9 [9]}}