We know that each installment is $, which means that Lucy needs to pay this installment every year for 10 years.
You can express the installments as the geometric series
Where the first part
and the quotient
There are two different methods that can be used to find the answer to this question. One is to use the formula for the sum of a geometric series
letting and using a calculator to calculate what we get. The expression becomes
which turns out to be the same as the size of the loan. That means you have shown that the annual interest is %, because when , the sum of the first 10 terms of the series is equal to the loan.
The second method is to set up an equation where the sum is equal to the size of the loan, and then solve the equation for the interest . The equation would look like this:
The easiest way to solve this is by using a digital tool, like
GeoGebra. That gives us two solutions,
but as the interest can’t be negative, we know that only can be a valid answer. That is also the value we wanted to show that the interest had, which means we have shown that the interest is %.
The bank claims that if a fund has an annual return of , then Lucy will gain a solid profit by investing in that fund.
Part 2 – Determine the value of Lucy’s money at the end of the year in the fund.
Lucy has invested all $ she loaned in the fund. With a guaranteed annual profit of % over a period of 10 years, you can use the formula for future value to find how much her money will be worth then. The formula is
where , and . That means the value of the money in the fund is
years, which means the value of Lucy’s money in the fund is $
Lucy’s net profit after 10 years is the difference between what she has paid for the loan and the value of her money in the fund.
Part 3 – Show that her net profit after 10 years will be .
You know that the future value of Lucy’s money in the fund is approximately $, but what’s the future value of the loan after years? You know that the present value of the loan is $, with an interest of %. You can use the formula again to find the future value of the loan:
Where , and . That makes the expression for the future value into
which means the future value of the loan in
years is approximately $
Lucy’s net profit is then
Instead of taking up this loan to invest in a fund, Lucy considers saving them in the bank. At the end of each year, she will deposit into an account with a fixed annual interest. She will make the first deposit in a year.
Part 4 – What does the interest on Lucy’s savings account need to be for her to have as much money in the bank after 10 years as in part 2?
The value of Lucy’s money in the fund at the end of the year was found to be $ in part 2.
Lucy’s savings can be expressed as the geometric series
where you can see that the first term of the series
and the quotient
The problem can be expressed as the equation
or by using the formula for the sum of a geometric series:
To solve this equation, it’s best to use a digital tool like
GeoGebra. That gives you
which means the interest on the savings account needs to be at least % for Lucy to have the same profit from it after years as she would get from investing in the fund, provided she deposits $ into the account each year.