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Date	November 2016	Marks available	2	Reference code	16N.1.SL.TZ0.T_10
Level	Standard Level	Paper	Paper 1 (with calculator from previous syllabus)	Time zone	Time zone 0
Command term	Find	Question number	T_10	Adapted from	N/A

Question

A hydraulic hammer drives a metal post vertically into the ground by striking the top of the post. The distance that the post is driven into the ground, by the $n th$ $n{\text{th}}$ strike of the hammer, is $d_{n}$ ${d_n}$ .

The distances $d_{1}, d_{2}, d_{3} \dots, d_{n}$ ${d_1},{\text{ }}{d_2},{\text{ }}{d_3}{\text{ }} \ldots ,{\text{ }}{d_n}$ form a geometric sequence.

The distance that the post is driven into the ground by the first strike of the hammer, $d_{1}$ ${d_1}$ , is 64 cm.

The distance that the post is driven into the ground by the second strike of the hammer, $d_{2}$ ${d_2}$ , is 48 cm.

Find the value of the common ratio for this sequence.

[2]

Find the distance that the post is driven into the ground by the eighth strike of the hammer.

[2]

Find the total depth that the post has been driven into the ground after 10 strikes of the hammer.

[2]

Markscheme

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

$48 = 64 r$ $48 = 64r$ (M1)

Note: Award (M1) for correct substitution into geometric sequence formula.

$= 0.75 (\frac{3}{4}, \frac{48}{64})$ $= 0.75\left( {\frac{3}{4},{\text{ }}\frac{{48}}{{64}}} \right)$ (A1) (C2)

[2 marks]

$64 \times (0.75)^{7}$ $64 \times {(0.75)^7}$ (M1)

Note: Award (M1) for correct substitution into geometric sequence formula or list of eight values using their $r$ $r$ . Follow through from part (a), only if answer is positive.

$= 8.54 (cm) (8.54296 \dots cm)$ $= 8.54{\text{ }}({\text{cm}}){\text{ }}(8.54296 \ldots {\text{ cm}})$ (A1)(ft) (C2)

[2 marks]

$depth = \frac{64 (1 - {(0.75)}^{10})}{1 - 0.75}$ ${\text{depth}} = \frac{{64\left( {1 - {{(0.75)}^{10}}} \right)}}{{1 - 0.75}}$ (M1)

Note: Award (M1) for correct substitution into geometric series formula. Follow through from part (a), only if answer is positive.

$= 242 (cm) (241.583 \dots)$ $= 242({\text{cm}}){\text{ }}(241.583 \ldots )$ (A1)(ft) (C2)

[2 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.3—Geometric sequences and series

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