To find the future value of an ordinary annuity, we use the Future Value of an Ordinary Annuity formula:
[tex]\[ FV = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right) \][/tex]
Where:
- [tex]\( PMT \)[/tex] is the regular payment amount (\[tex]$2,400 in this case).
- \( r \) is the periodic interest rate.
- \( n \) is the total number of periods.
Now, let's break down the steps to solve this problem:
1. Determine the periodic interest rate (\( r \)):
The annual interest rate is given as 1.50%. Since the interest is compounded monthly, we convert this annual rate to a monthly rate by dividing by 12:
\[
r = \frac{1.50\%}{12} = \frac{1.50}{100 \times 12} = \frac{1.50}{1200} = 0.00125
\]
2. Calculate the total number of periods (\( n \)):
The annuity is paid monthly for 3 years. Hence, the number of periods is:
\[
n = 12 \, \text{periods/year} \times 3 \, \text{years} = 36 \, \text{periods}
\]
3. Substitute the values into the future value formula:
\[
FV = 2400 \times \left( \frac{(1 + 0.00125)^{36} - 1}{0.00125} \right)
\]
4. Calculate the expression inside the parentheses:
First, calculate \( (1 + 0.00125)^{36} \):
\[
(1 + 0.00125)^{36} \approx 1.046997
\]
Next, subtract 1 from the result:
\[
1.046997 - 1 = 0.046997
\]
Finally, divide by 0.00125:
\[
\frac{0.046997}{0.00125} \approx 37.5976
\]
5. Multiply by the payment amount \( PMT \):
\[
FV = 2400 \times 37.5976 \approx 88317.05334
\]
6. Round the result to the nearest cent:
\[
FV \approx 88317.05
\]
So, the future value of the ordinary annuity is \$[/tex]88,317.05.
