Arithmetic Sequences
Geometric Sequences
Sequences Shuffle
Financial Applications
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
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
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
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
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
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
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)
5
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
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
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