You loan over 20 years. The interest on the loan is . The remaining loan for the first four years are , , and . What is the remaining loan after 13 years, and how much do you pay in total interest?
The exercise tells you that the remaining loans can be expressed as a series, where every annual remaining loan is a term in the series.
You can then find the remaining loan in year
. As this is an arithmetic series, you can use the formula to find the
th term in an arithmetic series,
. To do this, the first thing you need to find is
Then you can find an expression for
To find the remaining loan after 13 years, you insert into the expression for :
To find the total paid interest for the whole loan, you can look at each individual remaining loan. If you find the interest cost for each remaining loan and add them together, you will find the total amount of interest you’ve paid. This gives you a new arithmetic series, where the interest for the first remaining loan is , the interest for the second remaining loan is and the interest of the th remaining loan is
That makes the new arithmetic series look like this:
As the loan is paid over 20 years, you need to find the cost over 20 terms. You can find this by using the formula for the sum of an arithmetic series:
You know that and , and you need to find
First, you can find :
Then you can find :
That makes the total amount of interest you have paid
The total interest on a loan of $ with % interest is $ when the loan is repaid over 20 years.
You are paying back a total of
over 20 years.