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QUESTION 3 For the sequence 4 2 0 —2 —4 ... the common difference is (A) 4 B) 2 C) =2 (D) -4
QUESTION 4 Which of the following graphs could be modelled using a geometric sequence? (A) © 30 25 20 t 15 10 5 - - (B) (D) 30 25 20 t 15 10 5 0 30 25 20 {15 10 - o 1 2 3 4 5 6
QUESTION 18 (4 marks) Exhibition organisers believe that the number of attendees increases each day as an arithmetic sequence. The organisers know that 353 people attended the first day and 439 people attended the third day. a) Determine the common difference. [2 marks] b) Use the result from 18a) to predict the number of people who will attend the sixth day. [2 marks]
QUESTION 26 (5 marks) A scientist observed that the population of a specific bird species is decreasing by 17% each year and that at the beginning of 2016 there were 483 birds. a) Use a geometric sequence to model the bird population. [2 marks] b) Using the model from 26a) predict the number of birds remaining at the beginning of 2021. [3 marks]
QUESTION 1 (5 marks) A water tank contains 12 500 L of water. The tap is accidently left on and the tank loses 135 L per minute. The tap 1s turned off when the tank has 5000 L of water left. Use a mathematical model to determine how long the tap was left on to the nearest minute.
QUESTION 9 A staircase is to be extended by installing » additional stairs. Each stair is 25 cm high. If the existing staircase reaches 1.2 m off the ground which rule models the total height the stairs will reach in centimetres? (A) t =25+(n-1)x1.45 B) 1 =12+(n-1)x25 (C) t =145+(n-1)x25 (D) 1 =25+(n-1)x120
QUESTION 14 A town’s current population of 15480 is predicted to grow steadily at an annual rate of 12%. The predicted population after 10 years is approximately (A) 48079 (B) 34056 (C) 18576 (D) 17338
QUESTION 20 (4 marks) A farmer bought a tractor for $45 100 at the start of 2012. It depreciates by $2700 each year. Identify and use a mathematical model to determine the value of the tractor at the start of 2021.
QUESTION 7 (6 marks) The table shows the total number of times a new song is played on a music service in the days following its first release. Number of days since first release Total number of times played (’000s) The songwriter is paid 0.175 cents every time their song is played and will be paid after 60 days. They predict that by that time they will be owed at least $1000. Given that the number of times the song is played is increasing exponentially evaluate the reasonableness of this prediction.
QUESTION 4 A swimmer has a weekly training routine to improve their fitness as modelled by the recursive function T =T +2 where T is the number of laps they swim in week n and 7} = 4. Which graph best represents the swimmer’s routine? (A) (B) 16 0 % 14 —o 0 - 2 = 12 o s g 38 o 5 5 . 5 2 Z s 4 zZ 2 0 0 2 4 6 8 10 12 14 16 2 3 -+ 5 6 Number of laps per week Number of weeks © 6 » (D) s 24 5 d § Gy 5 & o & < 5 O ; = o S S g 2 s = = / o Z = 3 1 /! > 0 ® 0 2 4 6 8 10 12 14 16 0 1 2 3 4 5 6 Number of laps per week Number of weeks
QUESTION 5 The table lists the number of books sold per month by an online bookstore. If the simple 3-point moving average in October is 54 what is the simple 3-point moving average in May? o [ weo [ v [ aor [y [ [ [ e [ sep [ o [ [ e 5w e 6w e 5w 0] 0w (A) 69 (B) 73 (C) 74 (D) 89
QUESTION 18 (4 marks) The number of songs on a person’s playlist n in each week since joining a music streaming service ¢ forms an arithmetic sequence as shown by the graph. 700 % 600 575 > % 500 = 0453 o 2 400 S ®33] «w 300 o _qné 200 e209 Z 100 eR7 0 0 1 2 3 4 5 6 Time (weeks) Use the arithmetic sequence to predict the number of songs on this person’s playlist 25 weeks after joining the streaming service.
QUESTION 6 (7 marks) The first three lines in a pattern have the equations given. Their slopes form the terms of one sequence and their y-intercepts form the terms of another sequence. Each sequence is either arithmetic or geometric. Line 1: y=-0.8x + 1.2 Line2: y=0.4x+2.7 Line3: y=-0.2x+ 4.2 Determine the coordinates of the point where Line 5 in the pattern intersects Line 1.
QUESTION 6 In January 2022 40 fish were released into a new dam that has the capacity to support 10 000 fish. It is predicted that the dam will reach its capacity in January 2030 if the fish population doubles every year. Which sequence rule models the prediction? (A) ¢ =tr° ) where t = 40 r=2 n=8 (B) ¢ =t r° Y where t = 40 r= 2 n=9 (C) t =t +(n—l)d where t; = 40 d=2 n=8 (D) ¢ =t +(n—l)d where t = 40 d=2 n=9
QUESTION 9 Determine the 6th term of the arithmetic sequence that begins 3 9 ... (A) 21 (B) 33 (C) 45 (D) 729
QUESTION 15 The resale value of a boat shows geometric decay. Resale value ($) 50000 40000 32 000 Determine the resale value four years after purchase. (A) $16 000 (B) $20 480 (C) $22 000 (D) $25 600
QUESTION 5 (7 marks) At 9:00 am a security guard begins their patrol of the eight work sites represented in the network diagram starting and ending at site A. They drive at 40 km/h on the roads between sites and check every site once for 15 minutes. The length (km) of each road corresponds to the terms of the arithmetic sequence ¢ = ¢ + 2(n — 1) where ¢ = 1. Determine the earliest possible time the security guard can finish their patrol and identify the route they must follow. B ty C
QUESTION 4 A sequence is defined by aAn =0a -1 + 14 Given that as = 56 the value of a is A) 14 (B) 28 ©) 42 (D) 70
QUESTION 10 The fourth fifth and sixth terms of a geometric sequence are 7.5 3.75 and 1.875 respectively. The sum of the first and second terms in the geometric sequence is A) 15 (B) 30 (C) 45 (D) 90
QUESTION 22 (7 marks) A sequence has been defined as t 4 =t +4 t =5 a) Sketch the first six terms of the sequence in graphical form using the grid below. [3 marks] b) Use the rule t = t; + (n — 1)d to model the sequence in simplified form. [2 marks] c) Use the results from 22b) to calculate the 27th term. [2 marks]
QUESTION 27 (3 marks) Use the recursive rule t .q; = 0.65t t; = 120 to complete the following. a) Calculate the percentage by which each term decreases. [1 mark] b) Calculate t4. [2 marks]
QUESTION 7 (7 marks) In 2011 Chris was earning a salary of $67 500. Chris receives the same pay increase each year and in 2018 was earning a salary of $84 300. In 2015 Sam was earning a salary of $79 500. Sam receives the same pay increase each year and in 2019 was earning a salary of $87 900. Determine when Chris will earn more than Sam.
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