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Date	November 2020	Marks available	2	Reference code	20N.2.SL.TZ0.S_6
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	S_6	Adapted from	N/A

Question

An infinite geometric series has first term $u_{1} = a$ $u_{1} = a$ and second term $u_{2} = \frac{1}{4} a^{2} - 3 a$ $u_{2} = \frac{1}{4} a^{2} - 3 a$ , where $a > 0$ $a > 0$ .

Find the common ratio in terms of $a$ $a$ .

[2]

Find the values of $a$ $a$ for which the sum to infinity of the series exists.

[3]

Find the value of $a$ $a$ when $S_{\infty} = 76$ $S_{\infty} = 76$ .

[3]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

evidence of dividing terms (in any order) (M1)

eg $\frac{u_{1}}{u_{2}}, \frac{\frac{1}{4} a^{2} - 3 a}{a}$ $\frac{u_{1}}{u_{2}}, \frac{\frac{1}{4} a^{2} - 3 a}{a}$

$r = \frac{1}{4} a - 3$ $r = \frac{1}{4} a - 3$ A1 N2

[2 marks]

recognizing $| r | < 1$ $|r| < 1$ (must be in terms of $a$ $a$ ) (M1)

eg $∣ ∣ \frac{1}{4} a - 3 ∣ ∣ < 1, - 1 \leq \frac{1}{4} a - 3 \leq 1, - 4 < a - 12 < 4$ $|\frac{1}{4} a - 3| < 1, - 1 \leq \frac{1}{4} a - 3 \leq 1, - 4 < a - 12 < 4$

$8 < a < 16$ $8 < a < 16$ A2 N3

[3 marks]

correct equation (A1)

eg $\frac{a}{1 - (\frac{1}{4} a - 3)} = 76, a = 76 (4 - \frac{1}{4} a)$ $\frac{a}{1 - (\frac{1}{4} a - 3)} = 76, a = 76 (4 - \frac{1}{4} a)$

$a = \frac{76}{5} (= 15.2)$ $a = \frac{76}{5} (= 15.2)$ (exact) A2 N3

[3 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.8—Sum of infinite geo sequence

17N.1.SL.TZ0.S_10b:
The following diagram shows [CD], with length $b cm$ $b{\text{ cm}}$ , where $b > 1$ $b > 1$ . Squares with side lengths $k cm, k^{2} cm, k^{3} cm, \dots$ $k{\text{ cm}},{\text{ }}{k^2}{\text{ cm}},{\text{ }}{k^3}{\text{ cm}},{\text{ }} \ldots$ , where $0 < k < 1$ $0 < k < 1$ , are drawn along [CD]. This process is carried on indefinitely. The diagram shows the first three squares.

The total sum of the areas of all the squares is $\frac{9}{16}$ $\frac{9}{{16}}$ . Find the value of $b$ $b$ .
18M.1.SL.TZ1.S_10c:
Find the values of θ which give the greatest value of the sum.
22M.1.SL.TZ1.8a.ii:
Given that $p > 0$ $p > 0$ and $S_{\infty} = 3 + \sqrt{3}$ $S_{\infty} = 3 + \sqrt{3}$ , find the value of $x$ $x$ .
22M.1.AHL.TZ1.10a.ii:
Hence or otherwise, show that the series is convergent.
22M.1.AHL.TZ1.10a.iii:
Given that $p > 0$ $p > 0$ and $S_{\infty} = 3 + \sqrt{3}$ $S_{\infty} = 3 + \sqrt{3}$ , find the value of $x$ $x$ .
20N.2.SL.TZ0.S_6b:
Find the values of $a$ $a$ for which the sum to infinity of the series exists.
18M.1.SL.TZ1.S_10a.i:
Find an expression for r in terms of θ.
21M.2.SL.TZ1.7c.i:
Show that Chris will never reach the target if his initial deposit is $$ 9000$ $$ 9000$ .
20N.2.SL.TZ0.S_6c:
Find the value of $a$ $a$ when $S_{\infty} = 76$ $S_{\infty} = 76$ .
EXN.2.AHL.TZ0.9:
A biased coin is weighted such that the probability, $p$ $p$ , of obtaining a tail is $0.6$ $0.6$ . The coin is tossed repeatedly and independently until a tail is obtained.

Let $E$ $E$ be the event “obtaining the first tail on an even numbered toss”.

Find $P (E)$ $P (E)$ .
22M.1.AHL.TZ1.10a.i:
Show that $p = \pm \frac{1}{\sqrt{3}}$ $p = \pm \frac{1}{\sqrt{3}}$ .
22M.1.SL.TZ1.8a.i:
Show that $p = \pm \frac{1}{\sqrt{3}}$ $p = \pm \frac{1}{\sqrt{3}}$ .
21N.2.SL.TZ0.6a:
Find the first term of the sequence, $u_{1}$ $u_{1}$ .
21N.2.SL.TZ0.6b:
Find $S_{\infty}$ $S_{\infty}$ .
21N.2.SL.TZ0.6c:
Find the least value of $n$ $n$ such that $S_{\infty} - S_{n} < 0.001$ $S_{\infty} - S_{n} < 0.001$ .

Topic 1—Number and algebra

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Markscheme

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