11 General: Sequences & Finance Review

Arithmetic Sequences

1

Find the value of t₀ and D in the following arithmetic sequence: 14, 23, 32, 41, ...

t₀ = 14, D = 9

2

State the recurrence relation for the sequence 2, 8, 14, 20, ...

t_n+1 = t_n + 6,    t₀ = 2

3

Find the rule which describes the sequences -15, -7, 1, 9, ...

t_n = -15 + 8n

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.

t_n = t₀ + nD

t_n = 100 - 7n

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?

t_n+1 = t_n - 15, t₀ = 320

8 weeks until only 200 glasses left.

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

Geometric Sequences

1

Find the common ratio of the sequence 16, 24, 36, 54, ...

R = 1.5

2

Find the recurrence relation for the sequence 7, 14, 28, ...

t₀ = 7, t_n+1 = 2t_n

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.

V_n = 1200 x 0.7ⁿ

4

The recurrence relation for a sequence is t₀ = 30, t_n+1 = 1.1t_n. What is the value of the 15^th term in this sequence (to 2 decimal places)?

t₁₄ = 15^th term, so t₁₄ = 30 x 1.1¹⁴ = 113.92

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

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% 

Sequences Shuffle

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.

2

In the sequence given by the recurrence relation: t₀ = 63, t_n+1 = t_n-7, what is the value of t₁₅?

t₁₅ = -42

3

Express the following recurrence relation as a rule:

t₀ = 100, t_n+1 =  1/10 t_n

t_n = 100 x  (1/10)^n OR t_n =  100/10^n  OR t_n = 100 x 0.1ⁿ

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).

5

The following recurrence relation can be used to model the reducing balance after n months:

V₀ = 520,000, V_n+1 = 0.9965V_n

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)

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, t_n+1 = 2t_n + 4

Financial Applications

1

A tractor costs $90,000 when new. Its value depreciates at a flat rate of 10% per year. Let V_n 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.

V_n+1 = V_n - 9000, V₀ = 90,000

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.

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

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

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

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

Unit 1 Review

1

If sin(theta)=0.7886, then find the angle  theta to one decimal place.

theta=52.1^o

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 equation of a straight line that passes through the points (3, 4) and (6, 16)

y = 4x - 8

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

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

Find the volume of the composite object in the diagram to the nearest whole number.

V_TOTAL = V_{Hexagonal prism} + V_Pyramid

= A_base x height +  1/3 xx Area of base xxheight 

= 1056cm³