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Financial Maths, 2ADV M1 2023 HSC 25

On the first day of November, Jia deposits $10 000 into a new account which earns 0.4% interest per month, compounded monthly. At the end of each month, after the interest is added to the account, Jia intends to withdraw from the account.

Let be the amount (in dollars) in Jia's account at the end of months.

  1. Show that  .  (1 mark)

  2. Show that  .  (3 marks)

  3. Jia wants to be able to make at least 100 withdrawals.
  4. What is the largest value of that will enable Jia to do this?  (2 marks)

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a.   

 
   
   

 
b.   

♦ Mean mark (b) 44%.
 
   
 
   
   
   
   

 
c.   

 
 
 
 
   

 

Mean mark (c) 53%.

Financial Maths, 2ADV M1 2021 HSC 29

  1. On the day that Megan was born, her grandfather deposited $5000 into an account earning 3% per annum compounded annually. On each birthday after this, her grandfather deposited $1000 into the same account, making his final deposit on Megan’s 17th birthday. That is, a total of 18 deposits were made.
  2. Let be the amount in the account on Megan’s th birthday, after the deposit is made.
  3. Show that  .  (2 marks)

  4. On her 17th birthday, just after the final deposit is made, Megan has $30 025.83 in her account. You are NOT required to show this.
  5. Megan then decides to leave all the money in the same account continuing to earn interest at 3% per annum compounded annually. On her 18th birthday, and on each birthday after this, Megan withdraws $2000 from the account.
  6. How many withdrawals of $2000 will Megan be able to make?  (3 marks)

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a.   

 
 
 

 

b.   

 
 

 

 
 

 

Financial Maths, 2ADV M1 2020 HSC 26

Tina inherits $60 000 and invests it in an account earning interest at a rate of 0.5% per month. Each month, immediately after the interest has been paid, Tina withdraws $800.

The amount in the account immediately after the th withdrawal can be determined using the recurrence relation

,

where   and 

  1. Use the recurrence relation to find the amount of money in the account immediately after the third withdrawal.  (2 marks)

  2. Calculate the amount of interest earned in the first three months.  (2 marks)

  3. Calculate the amount of money in the account immediately after the 94th withdrawal.  (3 marks) 

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a.   
 
 

 

b.   


 


 

c.   

 
 
 
 
 

Financial Maths, 2ADV M1 2019 HSC 16a

A person wins $1 000 000 in a competition and decides to invest this money in an account that earns interest at 6% per annum compounded quarterly. The person decides to withdraw $80 000 from this account at the end of every fourth quarter. Let    be the amount remaining in the account after the th withdrawal.

  1.  Show that the amount remaining in the account after the withdrawal at the end of the eighth quarter is

     

    .  (2 marks)

  2.  For how many years can the full amount of $80 000 be withdrawn?  (3 marks)

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i. 

 
 
   
   
   

 

ii.   
   
 
   
   

♦♦♦ Mean mark part (ii) 19%.

 

 

Financial Maths, 2ADV M1 2018 HSC 16c

Kara deposits an amount of $300 000 into an account which pays compound interest of 4% per annum, added to the account at the end of each year. Immediately after the interest is added, Kara makes a withdrawal for expenses for the coming year. The first withdrawal is . Each subsequent withdrawal is 5% greater than the previous one.

Let    be the amount in the account after the th withdrawal.

  1. Show that  .  (1 mark)

  2. Show that  .  (1 mark)

  3. Show that there will be money in the account when
     
      (3 marks)

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♦ Mean mark 47%.

i.  
 
   
   

 

♦ Mean mark part (ii) 30%.

ii.  
   
   

 

iii.  
   
 
   

 

♦♦♦ Mean mark part (iii) 8%.

Financial Maths, 2ADV M1 2008 HSC 9b

Peter retires with a lump sum of $100 000. The money is invested in a fund which pays interest each month at a rate of 6% per annum, and Peter receives a fixed monthly payment from the fund. Thus the amount left in the fund after the first monthly payment is   .

  1. Find a formula for the amount, , left in the fund after monthly payments.   (2 marks)

  2. Peter chooses the value of so that there will be nothing left in the fund at the end of the 12th year (after 144 payments). Find the value of .   (3 marks)

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  1.  
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i.   

MARKER’S COMMENT: Students who developed this answer from writing the calculations of , , to then generalising for were the most successful.
 
 
 
 

 
 
 

 

ii.  

 

 

Financial Maths, 2ADV M1 2010 HSC 9a

  1. When Chris started a new job, $500 was deposited into his superannuation fund at the beginning of each month. The money was invested at 0.5% per month, compounded monthly. 

     

    Let    be the value of the investment after 240 months, when Chris retires.

     

    Show that      (2 marks)

  2. After retirement, Chris withdraws $2000 from the account at the end of each month, without making any further deposits. The account continues to earn interest at 0.5% per month.

     

    Let    be the amount left in the account    months after Chris's retirement.

     

      (1)  Show that  .     (3 marks)

     

      (2)  For how many months after retirement will there be money left in the account?     (2 marks)

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  2. (1) 
  3. (2) 
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i.     
 

 

MARKER’S COMMENT: Common errors included using , and taking the first term of the GP as 1 instead of 1.005 (note that the $500 goes in at the start of the month and earns interest before it is included in ).
 
 
 

 

 

ii. (1) 

IMPORTANT: At the end of the month, earns interest for the month BEFORE any withdrawal is made. Many students mistakenly had .
 
 
 
 

 
 
 

 

ii. (2)  

♦ Mean mark 38%

,