Arithmetic Sequences
Geometric Sequences
Sequences Shuffle
Financial Applications
Finance Solver
Unit 1 Review
1
Find the value of t0 and D in the following arithmetic sequence: 14, 23, 32, 41, ...
t0 = 14, D = 9
1
Find the common ratio of the sequence 16, 24, 36, 54, ...
R = 1.5
1
Is the following sequence arithmetic? Explain why or why not:
7, 15, 22, 30, ...
Not arithmetic: Does not increase by a common difference every time. E.g. 7 -> 15, d = 8, 15 -> 22, d = 7.
1
A tractor costs $90,000 when new. Its value depreciates at a flat rate of 10% per year. Let Vn be the value in dollars of the tractor after n years. Write a recurrence relation that models the depreciating value of this tractor over time.
Vn+1 = Vn - 9000, V0 = 90,000
1
The value of a reducing balance loan, in dollars, after n months, Vn, can be modelled by the recurrence relation:
V0 = 26,000; Vn+1=1.003 Vn - 400
What is the value of this loan after five months?
$24,380.31
1
If sin(theta)=0.7886, then find the angle theta to one decimal place.
theta=52.1^o
2
State the recurrence relation for the sequence 2, 8, 14, 20, ...
tn+1 = tn + 6, t0 = 2
2
Find the recurrence relation for the sequence 7, 14, 28, ...
t0 = 7, tn+1 = 2tn
2
In the sequence given by the recurrence relation: t0 = 63, tn+1 = tn-7, what is the value of t15?
t15 = -42
2
A computer initially costs $3000, but depreciates at a flat rate of 12% per year. How much does the computer depreciate by each year?
12% of $3000 = $360
The computer depreciates by $360 each year.
2
You take out a home loan of $450,000 at a rate of 4.35% per annum, compounding monthly. How much is owed to the bank after 30 years?
$1,655,549.03
2
A Pearson's correlation coefficient of 0.83 is found when testing the association between two variables. Interpret this in terms of strength, direction and shape.
This indicates that there is a strong, positive linear association between the variables.
3
Find the rule which describes the sequences -15, -7, 1, 9, ...
tn = -15 + 8n
3
Create a rule which models the depreciation of the value of a mobile phone which initially cost $1200 and loses 30% of its current value each year.
Vn = 1200 x 0.7n
3
Express the following recurrence relation as a rule:
t0 = 100, tn+1 = 1/10 tn
tn = 100 x (1/10)^n OR tn = 100/10^n OR tn = 100 x 0.1n
3
If the price of a 1L carton of milk is predicted to increase in price by 2.4% each year and currently costs $1.60, how much will it be worth after 20 years?
V20 = 1.60 x 1.02420 = $2.57
3
Samuel invests $500,000 in an annuity from which he receives a regular monthly payment. The balance of the annuity, in dollars, after n months, Vn, can be modelled by a recurrence relation of the form:
V0 = 500,000 Vn+1 = R Vn - 2000
For what value of R would this investment act as a perpetuity?
I = 4.8%, so R=1+(r/c)/100=1+(4.8/12)/100
or solve(500,000 = R x 500,000 - 2000, R))
R = 1.004
3
Find the equation of a straight line that passes through the points (3, 4) and (6, 16)
y = 4x - 8
4
A sequence started at 100 and had 7 subtracted each time to make new terms. Write a rule for finding the nth term in this sequence.
tn = t0 + nD
tn = 100 - 7n
4
The recurrence relation for a sequence is t0 = 30, tn+1 = 1.1tn. What is the value of the 15th term in this sequence (to 2 decimal places)?
t14 = 15th term, so t14 = 30 x 1.114 = 113.92
4
Is 67 a term in the sequence? If so, which term is it?
-5, 7, 19, ...
n = 7
Yes, it is the 7th term (t6).
4
The following recurrence relation can be used to model the reducing balance after n months:
V0 = 520,000, Vn+1 = 0.9965Vn
How many years will it take for the balance to be less than $400,000?
n = 74.83, so it will take 75 months (6 years and 3 months or 6.25 years)
4
Joseph borrowed $50,000 to buy a new car. Interest on this loan is charged at the rate of 7.5% per annum, compounding monthly. Joseph will fully repay this loan with 60 monthly payments over 5 years. If the final repayment is only $995.49, how much are the regular monthly repayments?
$1001.90
4
The relationship between two variables x and y as shown in the scatterplot is non-linear. Which transformation(s) will linearise the relationship?
y^2 ,log(x), 1/x
5
Initially, Fumbles Restaurant had 320 wine glasses. After one week, they only had 305 wine glasses. On average, 15 glasses are broken each week.
Write a recurrence relation for this scenario, and use it to determine how many weeks it takes at that breakage rate for there to be only 200 glasses left?
tn+1 = tn - 15, t0 = 320
8 weeks until only 200 glasses left.
5
The growth in covid-19 cases during the pandemic was found to be exponential, with confirmed cases for one country as follows: 25, 35, 49, ...
Based on this data, predict how many cases this country would have after 31 days (t30) to the nearest thousand.
t30 = 25 x 1.430 = 605,035.81 ~ 605,000 cases expected
5
You make a New Year's resolution to run every day for the next year. You set yourself the goal of running 1% further each day than you did the previous day. If you can run 1 km on the first day of the year, and increase this distance by 1% each day, write a rule to model how far you can run n days into the year, and use this to find how far will you be able to run on the final day of the year (round to 3 sig figs).
Rule: tn = 1.01n x 1
t365 = 1.01365 x 1 = 37.8 km
5
How long will it take for an investment to earn over $5000 interest when $14,500 is invested at 4.8% per annum with simple interest?
4.8% of $14500 = $696
5000 / 696 = 7.18, so 8 years
5
Ken has borrowed $70,000 to buy a new caravan. He will be charged interest at a rate of 6.9% per annum, compounding monthly. Ken will make monthly repayments of $800.
Find the total amount of interest Ken will have paid after 12 months.
He has paid: 12 x 800 = $9600
The balance is now: $65076.22, so he has repaid $4923.78 of the principal.
Interest is therefore 9600 - 4923.78 = $4676.22
5
The distribution of the daily price of 1kg of tomatoes from a certain store is approximately symmetric and bell-shaped with a mean of $6.49 and a standard deviation of $0.80. What percentage of days are the cost of tomatoes between $4.89 and $8.09?
95%
6
In an arithmetic sequence, t3 = 10 and t7 = 18. Determine the value of t0
t7 = t3 + 4D, so 18 = 10 + 4D, so D = 2.
t3 = t0 + 3 x 2, so 10 = t0 + 6, t0 = 4
6
Ned invests $10,000 at a rate of 6.2% p.a. compounding monthly over 5 years. At what interest rate would Ned need to invest this money if he wants to earn the same amount after only 4 years?
Solve: 10000xx(1+(6.2divide12)/100)^60=10000xx(1+(rdivide12)/100)^48
r=7.7549987579%
6
Determine the recurrence relation for the following sequence: 4, 12, 28, 60, ...
Combination: (0, 4), (1, 12), (2, 28), (3, 60), etc. Linear line connecting these is y = 2x + 4, so relation is:
t0 = 4, tn+1 = 2tn + 4
6
You invest an amount at 4% p.a. compound interest, compounded daily. After 5 years, the investment is worth $6106.95. What was the original amount invested?
$5000
6
Sam has been thinking about buying a new car for several months. His current car is a 2014 Toyota Corolla, which is silver and has travelled approximately 178,000 km. He spends around $65 per week on petrol and $900 each year on insurance.
Sam finds a car advertised for $28,500. It has a five-star safety rating, a 1.8-litre engine and a boot capacity of 440 litres. Sam has $8,500 available as a deposit, so the bank agrees to lend him $20,000.
The bank charges 6.5% p.a., with interest calculated monthly. Sam makes monthly repayments of $475. The loan will run for 4 years, after which the balance will be $0.
Identify N, I%, PV, PMT, FV, PPY and CPY.
N = 48; I% = 6.5; PV = 20,000; PMT = -475; FV = 0; PPY/CPY = 12
6
Find the volume of the composite object in the diagram to the nearest whole number.
VTOTAL = VHexagonal prism + VPyramid
= Abase x height + 1/3 xx Area of base xxheight
= 1056cm3