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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, GEN2 2023 VCAA 5

Arthur borrowed $30 000 to buy a new motorcycle.

Interest on this loan is charged at the rate of 6.4% per annum, compounding quarterly.

Arthur will repay the loan in full with quarterly repayments over six years.

  1. How many repayments, in total, will Arthur make?  (1 mark)

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

  1. Showing recursive calculations, determine the balance of the loan after two quarters.
  2. Round your answer to the nearest cent.   (1 mark)

  3. The final repayment required will differ slightly from all the earlier repayments of $1515.18
  4. Determine the value of the final repayment.
  5. Round your answer to the nearest cent.   (1 mark)

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a.   

b.   

c.   

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a.   

  
b.   

Mean mark (b) 51%.
♦♦ Mean mark (c) 25%

c.    

 
 
 
 
 
 

 

Recursion and Finance, GEN1 2022 VCAA 18-19 MC

The balance of a loan, , in dollars, after months is modelled by the recurrence relation


 

Question 18

The balance of the loan first falls below $398 000 after how many months?

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

 
Question 19

With a small change to the final payment, the loan is expected to be repaid in full in

  1. 25 years.
  2. 26 years.
  3. 28 years.
  4. 29 years.
  5. 30 years.
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Show Worked Solution

 
 
 

  

 

  

 
 
 
 
 
 

 


♦ Mean mark (Q19) 47%.

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 8

 

Samuel has a reducing balance loan.

The first five lines of the amortisation table for Samuel’s loan are shown below.
 


 

Interest is calculated monthly and Samuel makes monthly payments of $1600.

Interest is charged on this loan at the rate of 3.6% per annum.

  1. Using the values in the amortisation table
    1. calculate the principal reduction associated with payment number 3.   (1 mark)
    2. Calculate the balance of the loan after payment number 4 is made.
    3. Round your answer to the nearest cent.   (1 mark)
  2. Let be the balance of Samuel’s loan after months.
  3. Write down a recurrence relation, in terms of   and  , that could be used to model the month-to-month balance of the loan.   (1 mark)

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  1.  i.
  2. ii.

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a.i.  
   
   

 

♦ Mean mark part a.ii. 39%.

a.ii.  
   

 

 

♦♦ Mean mark part b. 30%.

b.  
 

CORE, FUR2 2019 VCAA 9

Phil would like to purchase a block of land.

He will borrow $350 000 to make this purchase.

Interest on this loan will be charged at the rate of 4.9% per annum, compounding fortnightly.

After three years of equal fortnightly repayments, the balance of Phil’s loan will be $262 332.33.

  1. What is the value of each fortnightly repayment Phil will make?

     

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

  2. What is the total interest Phil will have paid after three years?

     

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

  3. Over the next four years of his loan, Phil will make monthly repayments of $3517.28 and will be charged interest at the rate of 4.8% per annum, compounding monthly.

     

    Let be the balance of the loan months after these changes apply.

     

     

    Write down a recurrence relation, in terms of  and , that could be used to model the balance of the loan over these four years.   (2 marks)

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a. 

 


  

b.  
   
   

  
c. 

CORE, FUR1 2017 VCAA 19-20 MC

Shirley would like to purchase a new home. She will establish a loan for $225 000 with interest charged at the rate of 3.6% per annum, compounding monthly.

Each month, Shirley will pay only the interest charged for that month.

Part 1

After three years, the amount that Shirley will owe is

  1.    $73 362
  2.  $170 752
  3.  $225 000
  4.  $239 605
  5.  $245 865

 
Part 2

Let be the value of Shirley’s loan, in dollars, after months.

A recurrence relation that models the value of is

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