To find the future value of an ordinary annuity, we use the formula:
[tex]\[ FV = PMT \times \left( \frac{(1 + r)^{nt} - 1}{r} \right), \][/tex]
where:
- [tex]\( PMT \)[/tex] is the payment amount per period ($2,000 in this case),
- [tex]\( r \)[/tex] is the interest rate per period,
- [tex]\( n \)[/tex] is the number of times compounding occurs per year (monthly, so [tex]\( n = 12 \)[/tex]),
- [tex]\( t \)[/tex] is the number of years (6 years).
### Step-by-step Solution:
1. Determine the total number of periods:
[tex]\[ n \times t = 12 \times 6 = 72 \text{ periods} \][/tex]
2. Convert the annual interest rate to the interest rate per period:
Given an annual interest rate of [tex]\( 1.20\% \)[/tex], first convert this to a decimal:
[tex]\[ \text{Annual interest rate} = \frac{1.20}{100} = 0.012 \][/tex]
Since the interest is compounded monthly, divide the annual rate by 12:
[tex]\[ r = \frac{0.012}{12} = 0.001 \][/tex]
3. Substitute the values into the future value formula:
[tex]\[ FV = 2000 \times \left( \frac{(1 + 0.001)^{72} - 1}{0.001} \right) \][/tex]
4. Calculate the future value:
- First, calculate [tex]\((1 + 0.001)^{72}\)[/tex]:
[tex]\[ 1 + 0.001 = 1.001 \][/tex]
[tex]\[ (1.001)^{72} \approx 1.074562 \][/tex] (approximate value for explanation)
- Then, subtract 1:
[tex]\[ 1.074562 - 1 = 0.074562 \][/tex]
- Divide by the interest rate per period:
[tex]\[ \frac{0.074562}{0.001} = 74.562 \][/tex]
- Finally, multiply by the payment amount:
[tex]\[ 2000 \times 74.562 = 149,124 \][/tex]
### Final result:
The future value of the ordinary annuity is approximately:
[tex]\[ \boxed{149,233.37} \][/tex]
(rounded to the nearest cent).
