Iain opens an account to help save for a house. He opens the account at the beginning of 2017 with an initial deposit of \$50\,000 that is compounded annually at a rate of 5.9\% per annum. He makes further deposits of \$3000 at the end of each year.
Write an expression for the amount in the account after:
1 year
2 years
3 years
The amount in the account after n years can be expressed as: 50\,000 \times \left(1.059\right)^{n} + 3000 \times \left(1.059\right)^{n - 1} + \ldots + 3000 \times \left(1.059\right)^{2} + 3000 \times 1.059 + 3000This can be written as 50\,000 \times \left(1.059\right)^{n}, plus the sum of a geometric sequence. Write an expression representing the sum of geometric series.
Hence, determine the total value of Iain's savings by the start of 2024, to the nearest dollar.
At the start of 2014, Pauline deposits \$5000 into an investment account. At the end of each quarter, she makes an extra deposit of \$700.
The table below shows the first few quarters of 2014. All values in the table are in dollars:
| Quarter | Opening Balance | Interest | Deposit | Closing Balance |
|---|---|---|---|---|
| \text{Jan - Mar} | 5000 | 200 | 700 | 5900 |
| \text{Apr - Jun} | 5900 | 236.00 | 700 | 6836.00 |
| \text{Jul - Sep} | 6836.00 | 273.44 | 700 | 7809.44 |
Find the quarterly interest rate.
Write an expression for the amount in the account at the end of the first quarter.
Hence, write an expression for the amount in the account at the end of the second quarter.
Hence, write a similar expression for the amount in the account at the end of the third quarter.
Write an expression for the amount in the investment account after n quarters.
Hence, determine the total amount in Pauline’s account at the beginning of 2016 to the nearest dollar.
To save up to buy a car, Laura opens a savings account that earns 6\% per annum compounded monthly.
She initially deposits \$1400 when she opens the account at the beginning of the month, and then deposits \$165 at the end of every month.
The amount in the account after n months can be expressed as the nth term of a geometric sequence plus the sum of a different geometric sequence.
Write an expression for the amount in the investment account after n months.
Hence, determine the amount Laura has saved after 3 years.
The table below shows the first few years of an investment with regular deposits:
| Beginning balance | Interest | Deposit | End Balance | |
|---|---|---|---|---|
| 1 | y | 150 | 700 | 5850 |
| 2 | 5850 | 175.50 | 700 | 6725.50 |
| 3 | 6725.50 | x | 700 | 7627.27 |
| 4 | 7627.27 | 228.82 | 700 | w |
Find the value of:
Write a recursive rule for V_n in terms of V_{n - 1} and an initial condition V_0, that gives the value of the account after n years.
The total amount of this investment is withdrawn after 20 years. Find the value of the investment when its is withdrawn.
Calculate the total interest earned over the 20 years.
Buzz invests \$3000 at the end of each year for 3 years in a savings account that pays 7\% p.a. with interest compounded annually.
Calculate the value of his first deposit at the end of the 3 years.
Calculate the final value of his second deposit.
Calculate the future value of the annuity.
Calculate the interest Buzz will earn on this annuity.
A deposit of \$2000 is made on June 1, 2005 into an investment account and a deposit of \$200 is made each year on May 31. The balance at the end of each 12-month period for this investment, where interest is compounded annually, is given by A_{n} = 1.05 A_{n - 1} + 200, A_0 = 2000
State the annual interest rate.
Calculate the balance on June 1, 2006.
Calculate the value of the investment on June 1, 2012.
David opens an account to help save for a house. He opens the account at the beginning of 2011 with an initial deposit of \$40\,000 that is compounded annually at a rate of 4.3\% per annum. He makes further deposits of \$2000 at the end of each year.
How much money is in the account at the end of the first year?
Write a recursive rule for V_{n} in terms of V_{n - 1} that gives the value of the account after n years and an initial condition V_{0}.
Calculate the total value of his savings at the beginning of 2019.
Hence, determine how much interest David has earned.
Hannah opens a savings account to motivate herself to save regularly. She opens the account at the start of September 2013 with the intention of making regular deposits of \$180at the end of each month. The interest rate for this account is 24\% per annum which compounds at the end of each month.
Find the monthly interest rate of this account.
If Hannah first invests \$4000 when she opens the account in September 2013, calculate the balance at the end of the first month.
Calculate the value of her savings account at the end of her second month of saving.
Write a recursive rule for V_{n} in terms of V_{n - 1} that gives the value of the account after n months and an initial investment V_{0}.
Find the total value of her savings account at the start of September 2021.
Xanthe initially deposits \$8000 into an investment account. At the end of each quarter, Xanthe makes an extra deposit of \$500.
The table below shows the first few quarters of 2015. All values in the table are in dollars:
| Balance at the beginning of quarter | Interest | Deposit | Balance at end of quarter | |
|---|---|---|---|---|
| Jan-Mar | 8000 | 320 | 500 | 8820 |
| Apr-Jun | 8820 | 352.80 | 500 | 9672.80 |
| Jul-Sep | 9672.80 | 386.91 | 500 | 10\,559.71 |
| Oct-Dec |
Calculate the quarterly interest rate.
Calculate the nominal annual interest rate.
Complete the row for the last quarter.
Write a recursive rule for V_n in terms of V_{n - 1} and an initial condition V_0, that gives the value of the account after n quarters.
Calculate the total value of her savings at the beginning of 2017.
The table below shows the first few months of an investment with regular deposits. All values are given in dollars:
| Year | Beginning balance | Interest | Deposit | End balance |
|---|---|---|---|---|
| \text{Jan 2012} | y | 200 | 300 | 40\,500 |
| \text{Feb 2012} | 40\,500 | 202.50 | 300 | 41\,002.50 |
| \text{Mar 2012} | 41\,002.50 | x | 300 | 41\,507.51 |
| \text{Apr 2012} | 41\,507.51 | 207.54 | 300 | 42\,015.05 |
Find the monthly interest rate.
Find the value of:
x
y
Write a recursive rule for V_{n} in terms of V_{n - 1} that gives the value of the account after n months and an initial condition V_{0}.
The total amount of this investment is withdrawn after 6 years. Find the value of the investment when it is withdrawn.
Calculate the total interest earned over these 6 years.
Mr Jones opened a bank account for his granddaughter Sarah on the day she was born, January 5, 2006. He deposited \$5000.
Mrs Jones, Sarah’s grandmother, also deposited money into this account on that day, and continues to do so by depositing \$300 every 3 months.
The balance at the end of each quarter for this investment, where interest is compounded quarterly, is given by A_{n} = 1.03 A_{n - 1} + 300, A_{0} = 5300.
State the quarterly interest rate.
State the nominal annual interest rate.
Calculate the balance on the day after Sarah's first birthday.
Calculate the balance on the day after Sarah's 11th birthday.
At the end of each month, Kenneth deposits \$2000 into a savings account to help him set money aside for bills. This savings account earns 12\% p.a. interest, compounded monthly.
The table below shows the first few months of 2014. All values in the table are in dollars:
| Monthly payment | Balance at the beginning of month | Interest | Balance at the end of month | |
|---|---|---|---|---|
| March | 2000 | 40\,000 | 400 | 42\,400 |
| April | 2000 | 42\,400 | 424.00 | 44\,824.00 |
| May | 2000 | 44\,824.00 | 448.24 | 47\,272.24 |
| June |
Complete the row for June.
Write a recursive rule for V_{n} in terms of V_{n - 1} that gives the value of the account after n months and an initial condition V_{0}.
Find the balance at the beginning of August 2015, assuming no withdrawals have been made.
Maria invests \$6000 at the end of each year for 4 years in an investment account that pays 5\% per annum with interest compounded monthly.
Calculate the value of her first deposit at the end of the 4 years.
Calculate the value of her second deposit at the end of the 4 years.
Calculate the value of her third deposit at the end of the 4 years.
What is the future value of the annuity?
Consider the two investment options below:
Deposit a lump-sum of \$33\,000 at a rate of 6\% p.a. compounded annually for 30 years.
Deposit \$1100 annually in an annuity, at a rate of 6\% p.a. compounded annually for 30 years.
After 30 years, how much will the lump-sum investment be worth?
After 30 years, how much will the annuity be worth?
Which of the options is the best choice? Explain your answer.
An instalment of \$1500 is invested in a superannuation scheme on 1st April each year for 5 years, beginning in 2005. The money earns interest at 8\% p.a., compounded annually.
Calculate the value of the first instalment on 31st March 2010.
Calculate the value of the second instalment on 31st March 2010.
Calculate the value of the third, fourth, and fifth instalments on 31st March 2010.
These 5 amounts form a geometric sequence.
For this geometric sequence, find:
The first term, a.
The common ratio, r.
The number of terms, n.
The sum of n terms.
Joshua makes contributions of \$1800 to his superannuation scheme on 1st April each year. The money earns compound interest at 10\% per annum.
Let A_{15} be the total value of the fund at the end of 15 years.
Find the value of the first instalment at the end of 15 years.
Find the value of the second instalment at the end of 14 years.
Find the value of the last contribution at the end of just one year.
Hence, write a geometric series for A_{15}.
Hence find A_{15}, the total amount of the fund at the end of 15 years.
Laura makes contributions of \$300 to her superannuation scheme on the first day of each month. The money earns interest at 9\% per annum, compounded monthly.
Let A_{24} be the total value of the fund at the end of 24 months.
Find the value of the first instalment at the end of 24 months.
Find the value of the second instalment at the end of 23 months.
Find the value of the last contribution, invested for just 1 month.
Hence, evaluate A_{24}.
To enable Victor to have some money when he retires, he decided to invest \$1000 at the beginning of each year for the next 20 years. The superannuation fund he joined pays 9.2\% p.a., compounded annually.
The day after Victor invested his tenth \$1000, he was retrenched and could no longer afford to keep investing \$1000 each year. Victor left his contributions and the interest they had earned in the account to earn further interest.
Find the value of the investment at the end of 10 years, the year Victor makes his final contribution.
Hence, calculate the total value of his investment at the end of 20 years.
When Craig started contributing to superannuation, he paid \$500 into the fund at the beginning of each financial year. His fund pays 8.7\% p.a., compounded annually.
Craig decided to increase his yearly investment to \$900 because inflation was increasing the cost of living. He made 4 investments of \$500, all the following investments were \$900 p.a.
Craig plans to retire in 15 years. Let A_{n} be the value of each deposit at the end of 15 years.
Write down the value the following at the end of 15 years:
A_{2}
A_{3}
A_{4}
A_{5} is the first deposit of \$900. What is its value when Craig retires?
By considering the two geometric series formed by Craig's deposits, find out how much his superannuation will be worth in 15 years.
Suresh is going to invest \$130 in a superannuation fund at the start of every month for the next 5 years. Interest in the fund is compounded monthly and initially the rate was 1\% per month. Just before Suresh made his 7th payment, the interest rate dropped to 0.7\% per month and continued at this rate for the remainder of his investments.
What is the value of Suresh's superannuation fund just before the 7th payment is made?
What will this amount grow to at the end of the 5 years?
By considering a geometric series with 54 terms, find the final value of the payments after the interest rate drop.
Hence, calculate the total value of his superannuation fund after 5 years.
A person invests \$13\,000 each year in a superannuation fund. Compound interest is paid at 12\% p.a. on the investment. The first payment is made on 1st January 2001 and the last payment is made on 1st January 2020.
How much did the person invest over the life of the fund?
Calculate the value of the investment on 2001 at the end of 20 years.
Find the total value of the fund when it is paid out on 1st January 2021.
The person wants to reach a total value of \$1\,300\,000 in superannuation.
Write an expression for A_{n}, the value of the investment after n years.
Show that the target is reached when 1.12^{n} \gt \dfrac{12}{1.12} + 1.
At the end of which year will the superannuation be worth \$1\,300\,000?
Alison has planned a holiday which she has decided to take in 3 years time. She has estimated that she will need \$8000 and plans to save a fixed amount each month.
If she invests her savings at the beginning of each month in an account with interest at 7.5\% p.a. compounded monthly, what is the least amount of money she needs to save each month to reach her target in 3 years?
Annie was born on the 1st January 2000. Her parents invest \$1450 on this day and on every birthday thereafter. The interest is paid at 8\% compounded annually. After completing her HSC, she decides to use the account to fund a gap year. She withdraws all the funds on 31st December 2017 (getting paid her interest for 2017).
Calculate the value of the investment on 31st December 2001.
How much does Annie collect on 31st December 2017?
In 2001, the school fees at a private girls' school are \$11\,200 per year. Each year, the fees rise by 6 \dfrac{1}{2} \% due to inflation.
Susan is sent to the school, starting in Year 7 in 2001. If she continues through to her HSC year, how much will her parents have paid the school over the 6 years?
Susan's younger sister is starting in Year 1 in 2001. How much will they spend on her school fees over the next 12 years if she goes through to her HSC?
A man about to turn 25 is getting married. He has decided to pay \$4800 each year on his birthday into a combination life insurance and superannuation scheme that pays 9\% compound interest per annum. If he dies before age 65, his wife will inherit the value of the insurance to that point. If he lives to age 65, the insurance company will pay out the value of the policy in full.
The man is in a dangerous job. What will be the payout if he dies just before he turns 30?
The man's father died of a heart attack just before age 50. Suppose that the man also dies of a heart attack just before age 50. How much will his wife inherit?
Calculate the amount that the insurance company will pay if the man survives to his 65th birthday.
The spreadsheet below shows the first month of an investment with regular deposits:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | \text{Initial Investment} | 20\,000 | |||
| 2 | \text{Annual Interest Rate} | 0.072 | |||
| 3 | \text{Monthly Deposit} | 400 | |||
| 4 | |||||
| 5 | |||||
| 6 | \text{Month} | \text{Beginning Balance} | \text{Interest} | \text{Deposit} | \text{End Balance} |
| 7 | 1 | 20\,000 | 120 | 400 | 20\,520 |
| 8 | |||||
| 9 | |||||
| 10 |
Calculate the monthly interest rate for this investment.
Write a formula for cell \text{B7} in terms of one or more cells.
Write a formula for cell \text{C7} in terms of one or more cells.
Write a formula for cell \text{E7} in terms of one or more cells.
Find the end balance for the 5th year.
Calculate the total interest earned over the 60 months.
The spreadsheet below shows the first year of an investment with regular deposits:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | \text{Year} | \text{Beginning balance} | \text{Interest} | \text{Deposit} | \text{End balance} |
| 2 | 1 | 9000 | 270 | 500 | 9770 |
| 3 | |||||
| 4 | |||||
| 5 | |||||
| \ldots |
Calculate the annual interest rate for this investment.
Write a formula for cell \text{B6} in terms of one or more cells.
Write a formula for cell \text{C3} in terms of one or more cells.
Write a formula for cell \text{E4} in terms of one or more cells.
Using a spreadsheet program on your computer, or calculating manually, determine the end balance for the 6th year.
Calculate the total interest earned over these 6 years.