Preview Activity 8.1.1.
Suppose you receive \(\dollar5000\) through an inheritance. You decide to invest this money into a fund that pays \(8\%\) annually, compounded monthly. That means that each month your investment earns \(\frac{0.08}{12} \cdot P\) additional dollars, where \(P\) is your principal balance at the start of the month. So in the first month your investment earns
\begin{equation*}
5000 \left(\frac{0.08}{12}\right)
\end{equation*}
or \(\dollar33.33\text{.}\) If you reinvest this money, you will then have \(\dollar5033.33\) in your account at the end of the first month. From this point on, assume that you reinvest all of the interest you earn.
(a)
How much interest will you earn in the second month? How much money will you have in your account at the end of the second month?
(b)
Complete the table below to determine the interest earned and total amount of money in this investment each month for one year.
| Month | Interest earned |
Total amount of money in the account |
| \(0\) | \(\dollar0.00\) | \(\dollar5000.00\) |
| \(1\) | \(\dollar33.33\) | \(\dollar5033.33\) |
| \(2\) | ||
| \(3\) | ||
| \(4\) | ||
| \(5\) | ||
| \(6\) | ||
| \(7\) | ||
| \(8\) | ||
| \(9\) | ||
| \(10\) | ||
| \(11\) | ||
| \(12\) |
(c)
As we will see later, the amount of money \(P_n\) in the account after month \(n\) is given by
\begin{equation*}
P_n = 5000\left(1+\frac{0.08}{12}\right)^{n}\text{.}
\end{equation*}
Use this formula to check your calculations in the table above. Then find the amount of money in the account after 5 years.
(d)
How many years will it be before the account has doubled in value to $10000?
