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Date	May 2018	Marks available	5	Reference code	18M.2.AHL.TZ1.H_7
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	H_7	Adapted from	N/A

Question

It is known that the number of fish in a given lake will decrease by 7% each year unless some new fish are added. At the end of each year, 250 new fish are added to the lake.

At the start of 2018, there are 2500 fish in the lake.

Show that there will be approximately 2645 fish in the lake at the start of 2020.

[3]

Find the approximate number of fish in the lake at the start of 2042.

[5]

Markscheme

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

EITHER

2019: 2500 × 0.93 + 250 = 2575 (M1)A1

2020: 2575 × 0.93 + 250 M1

2020: 2500 × 0.93² + 250(0.93 + 1) M1M1A1

Note: Award M1 for starting with 2500, M1 for multiplying by 0.93 and adding 250 twice. A1 for correct expression. Can be shown in recursive form.

THEN

(= 2644.75) = 2645 AG

[3 marks]

2020: 2500 × 0.93² + 250(0.93 + 1)
2042: 2500 × 0.93²⁴ + 250(0.93²³ + 0.93²² + … + 1) (M1)(A1)

$= 2500 \times {0.93^{24}} + 250\frac{{\left( {{{0.93}^{24}} - 1} \right)}}{{\left( {0.93 - 1} \right)}}$ (M1)(A1)

=3384 A1

Note: If recursive formula used, award M1 for u_n = 0.93 u_n₋₁ and u₀ or u₁ seen (can be awarded if seen in part (a)). Then award M1A1 for attempt to find u₂₄ or u₂₅ respectively (different term if other than 2500 used) (M1A0 if incorrect term is being found) and A2 for correct answer.

Note: Accept all answers that round to 3380.

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series

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Topic 1—Number and algebra

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