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Date	May 2022	Marks available	5	Reference code	22M.1.AHL.TZ2.7
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 2
Command term	Justify and Find	Question number	7	Adapted from	N/A

Question

The sum of an infinite geometric sequence is $9$ .

The first term is $4$ more than the second term.

Find the third term. Justify your answer.

Markscheme

METHOD 1

$\frac{u_{1}}{1 - r} = 9$ A1

therefore $u_{1} = 9 - 9 r$

$u_{1} = 4 + u_{1} r$ A1

substitute or solve graphically: M1

$9 - 9 r = 4 + (9 - 9 r) r$ OR $\frac{4}{{(1 - r)}^{2}} = 9$

$9 r^{2} - 18 r + 5 = 0$

$r = \frac{1}{3}$ or $r = \frac{5}{3}$

only $r = \frac{1}{3}$ is possible as the sum to infinity exists R1

then $u_{1} = 9 - (9 \times \frac{1}{3}) = 6$

$u_{3} = 6 \times {\frac{1}{3}}^{2} = \frac{2}{3}$ A1

METHOD 2

$\frac{u_{1}}{1 - r} = 9$ A1

$r = \frac{u_{1} - 4}{u_{1}}$ A1

attempt to solve M1

$\frac{u_{1}}{1 - (\frac{u_{1} - 4}{u_{1}})} = 9$

$\frac{u_{1}}{(\frac{4}{u_{1}})} = 9$

${(u_{1})}^{2} = 36$

$u_{1} = \pm 6$

attempting to solve both possible sequences

$6, 2, \dots$ or $- 6, - 10 \dots$

$r = \frac{1}{3}$ or $r = \frac{5}{3}$

only $r = \frac{1}{3}$ is possible as the sum to infinity exists R1

$u_{3} = 6 \times {(\frac{1}{3})}^{2} = \frac{2}{3}$ A1

[5 marks]

Examiners report

Many candidates submitted quite poor attempts at this question. Many managed to state the equation $u_{1} = 9 (1 - r)$ obtained by considering the sum to infinity but few managed to find the second equation $u_{1} (1 - r) = 4$ . Common errors in failing to obtain this equation were that “four more” meant multiplied by four or thinking that the second term was four more than the first term. Even those candidates who obtained both equations were often unable to solve them. Attempted solutions often filled the page with algebra going nowhere. Most of those candidates who actually found the third term correctly then failed to realize that there were two solutions to the equations, one of which had to be rejected. Consequently, the final “reasoning” mark was seldom awarded.

Syllabus sections

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

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Topic 1—Number and algebra » AHL 1.11—Sum of infinite geometric sequences

Topic 1—Number and algebra

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