6. Number Patterns and Recursions

Worksheet

1

A paver needs to pave a floor with an area of 800 square metres. He can pave 50 square metres a day.

a

Complete the table showing the area left to pave at the start of each day:

\text{Day} | \text{Area of floor left to pave}\left(\text{m}^2 \right) |
---|---|

1 | 800 |

2 | |

3 | |

4 | |

5 |

b

What type of change is this?

A

Linear growth

B

Exponential growth

C

Linear decay

D

Exponential decay

2

A new lounge suite depreciates by a constant amount each year and its value is modelled by the recurrence relation

V_{n + 1} = V_n - 700, V_0 = 4800

where V_n is the value of the lounge suite, in dollars, after n years.

a

State the initial cost of the lounge suite.

b

State the amount of value lost each year.

3

An investment pays simple interest annually and is modelled by the recurrence relation

V_n = V_{n - 1} + 2310, V_0 = 30\,000

where V_n is the value of the investment after n years.

a

State the initial amount invested.

b

State the amount of interest paid each year.

c

Calculate the annual interest rate, expressed as a percentage. Round your answer to one decimal place.

4

Each day, Kata withdraws \$3 from her bank account to spend on lunch. Before she withdraws this amount on January 1, she has \$3000 in her bank account.

a

After Kata withdraws \$3 on January 7, how much does she have left in her bank account?

b

How much does Kata have left in her bank account at the end of February 5?

c

Write a recursive rule for t_{n + 1} in terms of t_n which defines the amount Kata has in her account at the end of day n, and an initial condition for t_0.

5

Zuber is a taxi service that charges a \$1.50 pick-up fee and \$1.95 per kilometre of travel.

a

Find the total charge for a 10\text{ km} journey.

b

Write a recursive rule for T_n in terms of T_{n - 1} which defines the price of a n\text{ km} trip, and an initial condition for T_0.

6

The value of Belem Bank shares is decreasing by \$1.55 each day. At the beginning of today's trading, the shares are worth \$48.83.

a

Today is March 6. How much are they worth at the start of March 11?

b

Write a recursive rule for V_{n + 1} in terms of V_n which defines the the value of the shares at the end of day n, and an initial condition for V_0.

7

Skye is a salesperson and is paid a travel allowance of \$40 for each business trip she takes to demonstrate the use of the product she sells. Each week she is also paid a retainer of \$220.

a

Calculate the total amount she is paid in a week where she does three business trips.

b

Write a recursive rule for u_{n + 1} in terms of u_n which defines how much Skye is paid in one week where she makes n trips, and an initial condition for u_0.

c

Write a recursive rule for v_{n + 1} in terms of v_n which defines how much Skye is paid in two weeks where she makes n trips, and an initial condition for v_0.

8

A piece of jewellery appreciates in value by a constant amount each year and its value is modelled by the following recurrence relation

V_{n + 1} = V_n + 220, V_0 = 3000

where V_n is the value of the jewellery, in dollars, after n years.

a

State the initial value of the piece of jewellery.

b

By how much does it appreciate each year?

c

Write an explicit rule for V_n that gives the value of the piece of jewellery after n years.

d

Find the value of the investment after 12 years.

9

A motorbike depreciates in value by a constant amount each year and its value is modelled by the recurrence relation

V_n = V_{n - 1} - 1200, V_0 = 15\,000

where V_n is the value of the motorbike, in dollars, after n years.

a

State the purchase price of the motorbike.

b

At what rate is it decreasing in value each year?

c

Write an explicit rule for V_n that gives the value of the motorbike after n years.

d

Find the value of the motorbike after 9 years.

10

The balance of a savings account earning simple interest each year is given by the explicit rule V_n = 2200 + 300 \left(n - 1\right) where V_n is the balance after n years.

a

How much interest is the account earning each year?

b

Calculate the account balance after 1 year.

c

State the original investment amount.

d

Write a recursive rule for V_{n+1} in terms of V_{n}, and an initial condition V_0.

11

For a photo, the animals at an animal shelter have been arranged such that there are 10 animals in the front row and each subsequent row has 7 more animals than the row in front of it.

a

Write a recursive rule for a_{n + 1}, the number of animals in the \left(n + 1\right)th row, in terms of a_n and an initial condition for a_1.

b

Write a general formula for a_n in terms of n.

c

How many animals are in the 7th row?

12

Maddox has just accepted an offer to play for the Budapest Football Club. He will initially be paid a salary of \$500\,000 a year, with this to increase by \$60\,000 each year.

Let t_n be Maddox's salary in the nth year.

a

State the value of t_1.

b

Write a recursive rule for t_{n + 1} in terms of t_n that describes Maddox's salary in a particular year.

c

Write a general formula for t_n in terms of n.

d

Hence, find Maddox's salary in year 8.

13

A diving vessel descends below the surface of the water at a constant rate so that the depth of the vessel after 4 minutes, 8 minutes and 12 minutes is 15 metres, 30 metres and 45 metres respectively.

If n is the number of minutes it takes to reach a depth of 120 metres, solve for n.

14

Tobias starts his career with a monthly wage of \$3500. At the beginning of each year that follows, he receives a raise and his monthly wage for that year will be \$160 greater than the previous year.

a

Find his yearly salary in the second year of his service.

b

Write down an expression for the total amount earned in m years.

c

Solve for the number of years, y, that it would take for him to earn a total of \$447\,120.

15

A car bought at the beginning of 2009 is worth \$1500 at the beginning of 2015. The value of the car has depreciated by a constant amount of \$50 each year since it was purchased.

a

Calculate the purchase price of the car in 2009.

b

Write an explicit rule for the value of the car after n years.

c

Solve for the year n at the end of which the car will be worth half the price it was bought for.

16

Farhan has saved up \$2540 to spend during his holiday to Lebanon. He plans to spend \$310 each week while he is there.

Let t_n be the amount of savings Farhan has left at the beginning of week n.

a

State the value of t_1.

b

Write a recursive rule for t_{n+1} in terms of t_{n} that describes how much Farhan has left to spend at the beginning of each week of his holiday.

c

For how many weeks will Farhan have enough money to spend during his holiday?

17

Megan is learning French using an app on her phone. Before using the app she already knows 30 words. The app gives her 10 new words to learn each day.

a

How many words will she know in total after using the app for 6 days?

b

Write a recursive rule, T_{n + 1}, the total number of words she has learnt after \left(n + 1\right) days of using the app, in terms of T_n.

c

Solve for n, the number of days it will take her to learn 500 words.

d

Write a recursive rule, W_{n + 1}, the total number of words she has learnt after \left(n + 1\right) weeks of using the app, in terms of W_n.

18

Marina is learning language using an app on her phone. Before using the app she already knows 40 words. The app gives her 6 new words to learn each day.

a

How many words will she know in total after using the app for 5 days?

b

Write a recursive rule for T_n, the total number of words she has learnt after n days of using the app, in terms of T_{n - 1}.

c

Solve for n, the number of days it will take her to learn 300 words.

d

State a recursive rule defining U_n, the total number words she will have learnt after n weeks of continuously using the app, in terms of U_{n - 1}.

19

When a new school first opened, a students started at the school. Each year, the number of students increases by the same amount, d.

a

At the beginning of its 7th year, it had 361 students. Form an equation for a in terms of d.

b

At the end of the 11th year, the school had 536 students. Form an equation for a in terms of d.

c

Hence, solve for d, the number of students who joined the school each year.

d

How many students started at the school when it first opened?

e

At the end of the nth year, the school has reached its capacity at 921 students. Solve for the value of n.

20

A mobile phone depreciates in value by a constant amount (in dollars) per month and its value is given by the explicit rule V_n = 800 - 25 n, where V_n is the value after n months.

a

By how much does the value of the phone depreciate each month?

b

State the purchase price of the phone.

c

Write a recursive rule for the value of the phone V_{n + 1}in terms of V_n, and an initial condition V_0.

21

A rare figurine was purchased for \$60 and ten years later it is worth \$460.

a

How much did the figurine appreciate by each year if it appreciated in value by a constant amount each year.

b

Write a recursive rule for V_n in terms of V_{n - 1}, and an initial condition V_0.

c

Write an explicit rule, V_n, for the value of the figurine after n years.

d

Find the value of the figurine in another 10 years time.

22

A piece of machinery depreciated at a constant rate per hour of use. After 160 hours of use, it was worth \$21\,040. After 210 hours of use, it was worth \$20\,740.

a

Determine the amount of depreciation each hour.

b

State V_0, the initial value of the machinery.

c

Write a recursive rule for V_{n + 1} in terms of V_n, and an initial condition V_0.

d

Write an explicit rule, V_n, for the value of the machinery after n hours of use.

e

Solve for n, the number of hours of use after which the machinery will be worth a quarter of its original value.

23

A piece of machinery depreciated at a constant rate per hour of use. After 140 hours of use, it was worth \$28\,300. After 190 hours of use, it was worth \$28\,050.

a

Determine the amount of depreciation each hour.

b

State V_0, the initial value of the machinery.

c

Write a recursive rule for V_n in terms of V_{n - 1}, and an initial condition V_0.

d

Write an explicit rule, V_n, for the value of the machinery after n hours of use.

e

Solve for n, the number of hours of use after which the machinery will be worth a quarter of its original value.

24

Glen decided to start saving by depositing some money in a safe each month.

On 1st January 2019, Glen put \$10 in the safe. On 1st February and 1st March in the same year, he deposited \$13 and \$16 respectively.

a

If he continued this pattern, how much did Glen put in the safe on August 1st 2019?

b

Write a recursive rule which gives the amount T_{n + 1} that Glen deposits in the safe \left(n + 1\right) months after January 1st 2019 in terms of T_n.

c

How much in total did Glen have in the safe by the end of 2019?

d

Glen wishes to save \$770. Solve for n, the number of months it will take to do this.

25

A cake is taken out of a 156 \degree \text{C} oven and left to cool in a room with a constant temperature of 26 \degree \text{C}. As soon as it is taken out, the temperature of the cake drops by 2.5 \degree \text{C} each minute.

a

Find the temperature of the cake after 8 minutes.

b

Write a recursive rule that gives the temperature T_n of the cake after n minutes in terms of T_{n - 1} with initial temperature T_0.

c

By how many degrees has the temperature of the cake dropped after half an hour?

d

Find the number of minutes it will take for the cake to reach room temperature.

26

A bobcat, initially purchased for \$70\,000, depreciates at a rate of \$12 for every day of use.

a

Write a recursive rule, V_{n + 1}, that gives the value of the bobcat, in dollars, after \left(n + 1\right) days of use.

b

How much value is lost over a five-week period, if the bobcat is used 6 days a week?

c

The owner of the bobcat will replace the tyres when the value drops below \$59\,209. After how many whole days of operation is he first due to change the tyres?

27

An investment of \$6000 pays simple interest at a rate of 4.2\% per annum and is modelled by the recurrence relation

V_n = V_{n - 1} + 252, V_0 = 6000

where V_n is the value of the investment after n years.

a

Calculate the value of the investment after 5 years.

b

How much interest has been earned in 5 years?

c

After how many whole years will the investment double?

28

An investment of \$5000 pays simple interest at a rate of 7.3\% per annum and is modelled by following the recurrence relation

V_{n + 1} = V_n + 365, V_0 = 5000

where V_n is the value of the investment after n years.

a

After how many whole years will the investment first be greater than \$17\,390?

b

After how many whole years will the investment first have earned more than \$16\,360 in interest?

29

An investment pays simple interest annually and is modelled by the recurrence relation

V_n = V_{n - 1} + 468, V_0 = 9000

where V_n is the value of the investment after n years.

a

State the initial amount invested.

b

State the amount of interest paid each year.

c

Calculate the annual interest rate.

d

After how many whole years will the investment double?

30

An investment of \$12\,000 earns \$700 in simple interest each year.

a

Write a recursive rule for V_n in terms of V_{n - 1} that gives the value of the investment after n years and an initial condition V_0.

b

Determine the value of the investment after 18 years.

c

After how many whole years will the value of the investment first exceed \$26\,900?

31

An investment of \$4000 earns simple interest at a rate of 4\% per annum.

a

Write a recursive rule for V_{n + 1} in terms of V_n that gives the value of the investment after n years and an initial condition V_0.

b

Determine the value of the investment after 11 years.

c

After how many whole years will the value of the investment first exceed \$8350?

32

A car is purchased for \$25\,000 and depreciates by \$1300 each year. The value of the car is modelled by

V_n = V_{n - 1} - 1300, V_0 = 25\,000

where V_n is the value of the car, in dollars, after n years.

a

Determine the value of the car after 1 year.

b

Determine the value of the car after 7 years.

c

By how much has the value of the car dropped after 7 years?

33

A motorbike purchased for \$5000 depreciates annually by \$300. The value of the motorbike is modelled by

V_{n + 1} = V_n - 300, V_0 = 5000

where V_n is the value of the motorbike, in dollars, after n years.

a

After how many whole years will the value of the motorbike first drop below \$1000?

b

After how many whole years will the motorbike first lose more than half its value?

34

A photocopier depreciates by a constant amount (in dollars) for every 100 pages printed. The value of the photocopier is modelled by the recurrence relation

V_n = V_{n - 1} - 11, V_0 = 2200

where V_n is the value, in dollars, of the photocopier after n hundreds of pages printed.

a

State the initial value of the photocopier.

b

State the constant rate of depreciation.

c

Determine the value of the photocopier after 2300 copies have been made.

d

The firm using the photocopier will replace it with a new one when its value drops below \$1780. After how many copies will this happen?

35

A car is initially purchased for \$24\,000 and depreciates by \$1700 each year.

a

Determine the value of the car after 7 years.

b

After how many whole years will the value of the car first fall below \$10\,100?

36

Julia has \$3000 to invest and deposits this amount into an account earning 7\% simple interest each year. Her sister, Lisa, has \$4000 to invest and deposits it into an account earning 4\% simple interest each year.

Find the number of whole years after which Julia's investment will be worth more than her sister's.

37

A courier van, initially purchased for \$49\,000, depreciates at a rate of \$180 for every 1000\text{ km} of use.

a

Determine the value of the van after it has travelled 11\,000\text{ km}.

b

After how many kilometres will the van first fall below \$47\,917 in value. Round your answer to the nearest multiple of 1000\text{ km} .

38

The value of a fridge depreciates by a constant amount each year and is modelled by the recurrence relation

V_{n + 1} = V_n - 300, V_0 = 1600

where V_n is the value of the fridge, in dollars, after n years.

Plot the value of the fridge for the first five years, including year 0 on a cartesian plane.

39

The value of an investment that pays simple interest each year is modelled by the recurrence relation

V_n = V_{n - 1} + 200, V_0 = 1000

where V_n is the value of the investment, in dollars, after n years.

a

Plot the value of the investment for the first five years, including year 0 on a cartesian plane.

b

Calculate the annual interest rate as a percentage.

40

A TV depreciates at a constant rate each year. The value, in dollars, of the TV after n years is represented by the graph shown.

a

State the initial value of the TV.

b

By how much does the value of the TV drop each year?

c

Write a recursive rule, V_{n + 1}, that gives the value of the TV after \left(n + 1\right) years and an initial condition V_0.

41

The value, in dollars, at the end of each year of an investment that pays simple interest annually is graphed.

a

State the initial value of the investment.

b

State the amount of interest paid each year.

c

Calculate the annual interest rate as a percentage.

d

Write a recursive rule for V_n in terms of V_{n - 1} that gives the value of the investment after n years and an initial condition V_0.

42

The value of an investment that pays simple interest each year is graphed, where V_n is the value of the investment, in dollars, after n years.

a

State the value of V_1.

b

State the value of d, the amount of interest earned each year.

c

Use the results from the previous parts to write an explicit rule for V_n.

d

Find the value of the investment after 18 years.

43

An investment pays simple interest annually. Its value, in dollars, at the end of each year has been drawn.

a

Write a recursive rule for V_{n + 1} in terms of V_n that gives the value of the investment after n years and an initial condition V_0.

b

Determine the value of the investment after 14 years.

c

How much interest has been earned after 14 years?

44

A tablet depreciates at a constant rate each year. The value, in dollars, of the tablet after n years is represented by the graph shown.

a

Write a recursive rule, V_{n + 1}, that gives the value of the tablet after \left(n + 1\right) years.

b

Determine after how many whole years the value of the tablet is first worth less than half of its initial value.

c

After how many whole years is the tablet considered worthless?

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Define and explain key concepts in use of a first-order linear recurrence relation to generate the terms of a number sequence, and apply a range of related mathematical routines and procedures

Define and explain key concepts in use of a recurrence relation to model and analyse practical situations involving discrete linear growth or decay such as a simple interest loan or investment, the depreciating value of an asset using the unit cost method; and the rule for the value of a quantity after n periods of linear growth or decay and its use, and apply a range of related mathematical routines and procedures