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Recursion and Finance, GEN2 2023 VCAA 7

Arthur takes out a new loan of $60 000 to pay for an overseas holiday.

Interest on this loan compounds weekly.

The balance of the loan, in dollars, after weeks, , can be determined using a recurrence relation of the form

  1. Show that the interest rate for this loan is 7.8% per annum.   (1 mark)

  2. Determine the value of in the recurrence relation if
    1. Arthur makes interest-only repayments   (1 mark)
    2. Arthur fully repays the loan in five years. Round your answer to the nearest cent.   (1 mark)
  3. Arthur decides that the value of will be 300 for the first year of repayments.  
  4. If Arthur fully repays the loan with exactly three more years of repayments, what new value of will apply for these three years? Round your answer to the nearest cent.   (1 mark)
  5. For what value of does the recurrence relation generate a geometric sequence?   (1 mark)

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a.   

b.i. 

b.ii.

c.   

d.   

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a.   



♦♦ Mean mark (a) 32%.



 
b.i. 
 

b.ii.

 
 
 
 
 
 

 

 



♦ Mean mark (b)(i) 47%.
♦ Mean mark (b)(ii) 40%.



c.   

 
 
 
 
 
 

 

 

 
 
 
 
 
 

 

 



♦♦ Mean mark (c) 32%.



d.    



♦♦♦ Mean mark (d) 11%.



Recursion and Finance, GEN1 2022 VCAA 23 MC

Li invests $4000 for five years at 3.88% per annum, compounding annually.

Joseph invests a sum of money for five years, which earns simple interest paid annually.

Let be the value, in dollars, of Joseph's investment after years.

The two investments will finish at the same value, rounded to the nearest cent, if Joseph's investment is modelled by which one of the following recurrence relations?

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♦♦ Mean mark 49%.

CORE, FUR2 2021 VCAA 8

For renovations to the coffee shop, Sienna took out a reducing balance loan of $570 000 with interest calculated fortnightly.

The balance of the loan, in dollars, after fortnights, can be modelled by the recurrence relation
 

                 
 

  1. Calculate the balance of this loan after the first fortnightly repayment is made.   (1 mark)

  2. Show that the compound interest rate for this loan is 2.6% per annum.   (1 mark)

  3. For the loan to be fully repaid, to the nearest cent, Sienna's final repayment will be a larger amount.
  4. Determine this final repayment amount.
  5. Round your answer to the nearest cent.   (1 mark)

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a.  
   


b. 


 

 

 

c. 

 

 

 

 

 

CORE, FUR2 2020 VCAA 9

Samuel opens a savings account.

Let be the balance of this savings account, in dollars, months after it was opened.

The month-to-month value of can be determined using the recurrence relation shown below.

  1. Write down the value of , the balance of the savings account after four months.
  2. Round your answer to the nearest cent.   (1 mark)

  3. Calculate the monthly interest rate percentage for Samuel’s savings account.   (1 mark)

  4. After one year, the balance of Samuel’s savings account, to the nearest dollar, is $5183.

     

    If Samuel had deposited an additional $50 at the end of each month immediately after the interest was added, how much extra money would be in the savings account after one year?

     

    Round your answer to the nearest dollar.   (1 mark)

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a.  
 

 

 

b. 

 

c.   

 

CORE, FUR1-NHT 2019 VCAA 19 MC

Consider the recurrence relation shown below.

This recurrence relation could be used to determine the value of

  1. a perpetuity with a payment of $2000 per quarter.
  2. an annuity with withdrawals of $2000 per quarter.
  3. an annuity investment with additional payments of $2000 per quarter.
  4. an item depreciating at a flat rate of 1.3% of the purchase price per quarter.
  5. a compound interest investment earning interest at the rate of 1.3% per annum.
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CORE, FUR2 2019 VCAA 8

Phil invests $200 000 in an annuity from which he receives a regular monthly payment.

The balance of the annuity, in dollars, after months, , can be modelled by the recurrence relation

  1. What monthly payment does Phil receive?   (1 mark)

  2. Show that the annual percentage compound interest rate for this annuity is 4.2%.   (1 mark)

At some point in the future, the annuity will have a balance that is lower than the monthly payment amount.

  1. What is the balance of the annuity when it first falls below the monthly payment amount?

     

    Round your answer to the nearest cent.   (1 mark)

  2. If the payment received each month by Phil had been a different amount, the investment would act as a simple perpetuity.

     

    What monthly payment could Phil have received from this perpetuity?   (1 mark)

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a. 

b.  
 

  
c. 

 

 

  

d. 

 

CORE, FUR2 2018 VCAA 5

After three years, Julie withdraws $14 000 from her account to purchase a car for her business.

For tax purposes, she plans to depreciate the value of her car using the reducing balance method.

The value of Julie’s car, in dollars, after years, , can be modelled by the recurrence relation shown below

  1. For each of the first three years of reducing balance depreciation, the value of is 0.85

     

    What is the annual rate of depreciation in the value of the car during these three years?  (1 mark)

  2. For the next five years of reducing balance depreciation, the annual rate of depreciation in the value of the car is changed to 8.6%.

     

    What is the value of the car eight years after it was purchased?
    Round your answer to the nearest cent.  (2 marks)

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a.  

COMMENT: Note almost half of students answered incorrectly here!

     

  
b.   

♦ Mean mark 42%.

 

 

 
 

CORE, FUR1 2018 VCAA 17-18 MC

The value of an annuity investment, in dollars, after years, , can be modelled by the recurrence relation shown below.

 
Part 1

What is the value of the regular payment added to the principal of this annuity investment?

  1.   $34.00
  2. $156.40
  3. $466.00
  4. $500.00
  5. $656.40

 
Part 2

Between the second and third years, the increase in the value of this investment is closest to

  1.   $656
  2.   $658
  3.   $661
  4. $1315
  5. $1975
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