Roald invests \$1700 each year for 25 years into an account with an interest rate of 4.5\% p.a., compounded annually. Determine the future value of this annuity at the end of the investment period.
To start saving money for retirement, Christa invests \$375 each quarter at an interest rate of 9\% p.a., compounded quarterly. Determine the future value of this annuity after 25 years.
Fred invests \$300 each quarter for 30 years into an account with an interest rate of 4\% p.a., compounded quarterly. Determine the future value of this annuity at the end of the investment period.
Regular semiannual payments with an interest rate of 1.5\% p.a., compounded semiannually for 10 years. Determine the minimum regular payment needed to accumulate to \$15\, 000.
Regular monthly payments with an interest rate of 3\% p.a., compounded monthly for 30 years. Determine the minimum regular payment needed to accumulate to \$280\, 000.
Mae is saving for a new car that costs \$22\, 000. To accumulate enough money to buy the car in 7 years, how much money should she invest each month in an annuity with a 4.5\% p.a. interest rate, compounded monthly?
The city of Bayfield has a debt of \$24\,000\,000. To accumulate enough money to repay the debt in 40 years, how much money should the city invest semiannually into an annuity with a 5.5\% p.a. interest rate, compounded semiannually?
Judy wants to go on a long overseas trip in 7 years time and calculates that she will need at least \$17\, 300.42 to pay for it. The bank Judy intends to use offers her an account which earns 12\% p.a. where interest is compounded monthly.
If she deposits one lump sum today and doesn't make any further contributions, what is the least amount Judy would need to deposit today so that she reaches her goal in time for her trip?
Judy realises that she might not have enough money to deposit today, so she decides to instead make regular, equal contributions every compounding period. Find the minimum amount Judy would need to deposit each compounding period so that she reaches her goal in time for her trip.
To save money for a masterβs degree, you deposit \$2280 at the end of each year in an annuity with a rate of 3\% p.a. compounded annually.
Find how much money you will have saved at the end of 5 years.
Find the total interest earned.
A different bank offers you an investment plan where the annual interest rate is the same, but interest is compounded monthly and you make monthly contributions of \$190. Under this plan, how much money would you have saved after 5 years?
To offer scholarship funds to children of employees, a company invests \$15\,000 at the end of every three month period in an annuity that pays 7\% p.a. compounded quarterly.
Find how much money the company will have in scholarship funds at the end of 8 years.
Find the total interest earned.
A different company wants to save the same amount of money at the same interest rate, but does not want to use an annuity. They want to make a deposit of a single sum with no subsequent deposits. How much would this company have to deposit to have the same amount after 8 years? Assume that interest is compounded every 3 months.
Arturo makes regular deposits of \$6000 at the end of each year into an account earning 5\% p.a. compounded annually.
If Arturo makes no initial deposit, write a recurrence relation for this situation, where A_n is the balance at the end of the nth year and A_0 is the initial investment.
Complete the following table, all values given are in dollars:
| Year | Balance at beginning of year | Interest | Annual payment | Balance at end of year |
|---|---|---|---|---|
| 1 | 0 | 0 | 6000 | 6000 |
| 2 | 6000 | 300 | 6000 | 12\,300 |
| 3 |
Find the total interest earned over the three years.
Find the value of the investment after 5 years.
Frida makes regular deposits of \$400 at the end of each month into an account earning 4.2\% p.a. compounded monthly.
If Frida makes no initial deposit, write a recurrence relation for this situation, where A_n is the balance at the end of the nth month and A_0 is the initial investment.
Complete the following table, all values given are in dollars:
| Month | Balance at beginning of month | Interest | Monthly payment | Balance at end of month |
|---|---|---|---|---|
| 1 | 0 | 0 | 400 | 400 |
| 2 | 400 | 1.40 | 400 | 801.40 |
| 3 |
Find the total interest earned over the three months.
Find the value of the investment after 2 years.
David makes regular deposits of \$500 at the end of each year into an account earning 4\% p.a. compounded annually.
If David makes no initial deposit, write a recurrence relation for this situation, where A_n is the balance at the end of the nth year and A_0 is the initial investment.
Find the value of the investment after 3 years.
Find the value of the investment after 10 years.
Michael makes regular deposits of \$200 at the end of the month into an account earning 3.6\% p.a. compounded monthly.
If Michael makes no initial deposit, write a recurrence relation for this situation, where A_n is the balance at the end of the nth month and A_0 is the initial investment.
Find the value of the investment after 5 months.
Find the value of the investment after 4 years.
Find the interest earned on the investment over 4 years.
Eileen makes regular deposits at the end of each month into an account earning 3.9\% p.a. compounded monthly.
Harry makes regular deposits at the end of each month into an account earning 2.4\% p.a. compounded monthly.
Christa invests \$5000 at 4.2\% p.a. compounded monthly and makes an additional monthly payment of \$200 at the end of each month.
Write a recurrence relation for this situation, where A_n is the balance at the end of the \\ nth month and A_0 is the initial investment.
Find the value of the investment after 2 months.
Determine how many whole months it takes for the balance to exceed \$7500.
James invests \$8000 at 2.4\% p.a. compounded monthly and makes an additional monthly payment of \$350 at the end of each month.
Write a recurrence relation for this situation, where A_n is the balance at the end of the nth month and A_0 is the initial investment.
Find the value of the investment after 4 months.
Determine how many whole months it takes for the balance to exceed \$12\ 000.
A deposit of \$3000 is made on June 1, 2006 into an investment account and a deposit of \$400 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} = 1.04 A_n + 400, \, A_0 = 3000State the annual interest rate.
Find the balance on June 1, 2007.
Find the value of the investment on June 1, 2014.
Mr Jones opened a bank account for his granddaughter Victoria on the day she was born, January 5, 2007. He deposited \$3000. Mrs Jones, Victoriaβs grandmother, also deposited \$500 into this account on that day, and continues to do so by depositing \$500 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} = 1.03 A_n + 500, \, A_0 = 3\,500
Find the nominal annual interest rate.
Calculate the balance on the day after Victoria's first birthday.
Calculate the balance on the day after Victoria's 12th birthday.
Lucy makes regular deposits of \$150 at the end of the month into an account earning 2.7\% p.a. compounded monthly.
If Lucy makes no initial deposit, write a recurrence relation for this situation, where A_n is the balance at the end of the nth month and A_0 is the initial investment.
Find the value of the investment after 4 months.
Find the value of the investment after 3 years.
If Lucy had also made an initial deposit of \$3000 at the beginning of the first month, what would be the value of the investment after three years?
Find the interest earned on the investment from part (d).
Abbey makes regular deposits of \$250 at the end of the month into an account earning 3.9\% p.a. compounded monthly.
If Abbey makes no initial deposit, write a recurrence relation for this situation, where A_n is the balance at the end of the nth month and A_0 is the initial investment.
Find the value of the investment after 3 months.
Find the value of the investment after 2 years.
If Abbey had made an initial deposit of \$2000 at the beginning of the first month, what would be the value of the investment after two years?
Find the interest earned on the investment from part (d).
Bill opens an account to help save for a house. He opens the account at the beginning of 2013 with an initial deposit of \$40\,000 that is compounded annually at a rate of 3.7\% per annum. He makes further deposits of \$1000 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} and an initial condition V_0, that gives the value of the account after n years.
Calculate the total value of his savings at the beginning of 2021.
Hence, determine how much interest Bill earned.
Sandy 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 \$110 at 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 Sandy first invests \$2000 when she opens the account and makes no further deposits in September 2013, find the balance at the end of the first month.
Find the value of her savings account at the end of her second month.
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.
Determine total value of her savings account at the start of September, 2019.
At the end of each month, Uther deposits \$2000 into a savings account to help him set money aside for bills. This savings account earns 24\% p.a. interest, compounded monthly.
The table below shows the first few months of 2015. All values in the table are in dollars:
| Balance at beginning of month | Interest | Monthly payment | Balance at end of month | |
|---|---|---|---|---|
| March | 20\,000 | 400 | 2000 | 22\,400 |
| April | 22\,400 | 448.00 | 2000 | 24\,848.00 |
| May | 24\,848.00 | 496.96 | 2000 | 27\,344.96 |
| June |
Complete the row for June.
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 months.
Determine the balance at the beginning of August 2016, assuming no withdrawals have been made.
Sarah initially deposits \$8000 into an investment account. At the end of each quarter Sarah makes an extra deposit of \$500.
The table below shows the first few quarters of 2011. All values in the table are in dollars:
| Balance at 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 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 2013.
The table below shows the first few years of an investment with regular deposits. All values given are in dollars:
| Year | Balance at beginning of year | Interest | Deposit | Balance at end of year |
|---|---|---|---|---|
| 1 | y | 150 | 700 | 5\,850 |
| 2 | 5\,850 | 175.50 | 700 | 6\,725.50 |
| 3 | 6\,725.50 | x | 700 | 7\,627.27 |
| 4 | 7\,627.27 | 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.
The table below shows the first few months of an investment with regular deposits. All values are given in dollars:
| Beginning balance | Interest | Deposit | End Balance | |
|---|---|---|---|---|
| Jan 2013 | y | 50 | 100 | 20\,150 |
| Feb 2013 | 20\,150 | 50.38 | 100 | 20\,300.38 |
| Mar 2013 | 20\,300.38 | x | 100 | 20\,451.13 |
| Apr 2013 | 20\,451.13 | 51.13 | 100 | 20\,602.26 |
Calculate the monthly interest rate.
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 months.
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 the 6 years.
Jenny is saving for a European holiday which will cost \$15\,000. She puts the \$5000 she has already saved in a savings account which offers interest compounded quarterly. She also makes a quarterly contribution of \$350. The progression of the investment is shown in the table below with some values missing, all values given are in dollars:
| Quarter | Balance at start of quarter | Interest | Payment | Balance at end of quarter |
|---|---|---|---|---|
| 1 | 5000 | 46.88 | 350 | 5396.88 |
| 2 | 5396.88 | 50.60 | 350 | 5797.48 |
| 3 | 5797.48 | 350 |
If Jenny changes the payment to \$400 per quarter, how much sooner will she be able to go on holidays?
Dave opened a savings account at the beginning of February 2012, where the interest is compounded monthly. His account balance at the beginning of March, April and May are shown in the table:
Write a recursive rule for V_n in terms of V_{n - 1} and an initial condition for V_1, that gives the balance in the account at the beginning of the nth month.
| Month | Balance |
|---|---|
| \text{March} | \$5100 |
| \text{April} | \$5202 |
| \text{May} | \$5306.04 |
Marly has a goal to save \$60\,000 for a deposit on a house in 4 yearsβ time. She has already saved \$5000 and plans to make regular monthly payments into an account earning 4.8\% p.a. compounded monthly.
Find the minimum monthly payment required to reach her goal.
Hakim has a goal to save \$10\,000 for a holiday in 2 yearsβ time. He has already saved \$3000 and plans to make regular quarterly payments into an account earning 3.6\% p.a. compounded monthly.
Find the minimum monthly payment required to reach his goal.
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 | 6\,000 | 660 | 500 | 7160 |
| 3 | |||||
| 4 | |||||
| 5 |
Calculate the annual interest rate for this investment.
Write a formula for cell \text{B3} in terms of one or more other cells.
Write a formula for cell \text{C6} in terms of \text{B6}.
Write a formula for cell E5 in terms of one or more other cells.
Using a spreadsheet program, reproduce this spreadsheet and determine the end balance for the 4th year.
Calculate the total interest earned over the 4 years.
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 |
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.
Using a spreadsheet program, reproduce this spreadsheet and determine the end balance for the 5th year.
Calculate the total interest earned over the 60 months.
Valerie is saving for a European holiday which will cost \$20\,000. She puts the \$8000 she has already saved in a savings account which offers 3.75\% p.a compounded quarterly. She also makes a quarterly contribution of \$400.
Use technology to calculate the number of whole quarters it will take for Valerie to save the \$20\,000 required for the holiday.
An investor deposits \$12\,000 into a high earning account with interest of 2.5\% p.a. compounded monthly and makes \$80 per month deposits into the account.
Use technology to calculate the number of whole months it will take until the investment doubles.
How much should the investor deposit each week if they want the original investment to triple at the end of three years?
Bill is saving for a skiing holiday which will cost \$3200. He puts the \$500 he has already saved into a savings account which offers 5.25\% p.a compounded monthly. He also makes a monthly contribution of \$100. Bill is interested in finding out how long it will take him to save up for his holiday.
Use technology to calculate the number of whole months it will take for Bill to save the \$3200 required for the holiday.
A student deposits \$600 into a special students only savings account advertising an interest rate of 4.95\% p.a. compounded weekly. She also makes \$20 per week deposits into the account.
Use technology to calculate the number of whole weeks it will take until the investment doubles. Assume there are 52 weeks in a year.
How much should the student deposit each week if they want the original investment to grow to \$2500 at the end of four years?
An investor deposits \$20\,000 into a high earning account with interest of 4.5\% p.a. compounded weekly and makes weekly deposits into the account.
How much should the investor deposit each week if they want the original investment to double at the end of three years?
Pauline and Yuri are saving for a house extension which will cost \$330\,000. They put the \$50\,000 they have already saved into an investment account which offers make a contribution of \$34 per day (which is approximately \$1000 per month).
Use technology to calculate the number of whole years it will take for Pauline and Yuri to save the \$330\,000 required for the extension.
A self-funded retiree deposits \$200\,000 into a special savings fund advertising an interest rate of 5.05\% p.a. compounded quarterly. An overseas pension also contributes \$360 per quarter into the account.
Use technology to calculate the number of whole quarters it will take until the investment grows by \$15\,000.
A business man deposits \$23\,500 into an investment account advertising an interest rate of 3.85\% p.a. compounded daily. He also makes \$10 per day deposits into the account.
Use technology to calculate the number of whole days it will take until the investment triples. Assume there are 365 days in a year.
How much should the business man deposit each day in dollars if they want the original investment to triple at the end of four years?
Kathleen aims to have \$250\,000 in her savings account by the time she is 33. She just turned 23 years old and has \$3325 in the account. The current interest rate is 3.2\% p.a. compounded monthly.
Use technology to calculate the monthly payment required for Kathleen to save the \$25\,000 by the time she is 33 years old. Assume there are 365 days per year.
Amelia and Ryan are saving for a new caravan which will cost \$120\,000. They put the \$40\,000 they have already saved into an investment account which offers 3.015\% p.a. compounded weekly. They also make a contribution of \$50 per week.
Use technology to calculate the number of whole weeks it will take for Amelia and Ryan to save the \$120\,000 required for the extension. Assume there are 52 weeks in a year.