Find the future values of the ordinary annuities at the given annual rate r compounded as indicated. The payments are made to coincide with the periods of compounding. (Round your answer to the nearest cent.)
PMT = $120, r = 2.3%, compounded weekly for 15 years

To calculate the future value $ FV $ of an ordinary annuity, we can use the formula:

\[
FV = PMT \times \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}}
\]

where:
- $ PMT $ is the periodic payment (in this case, $120),
- $ r $ is the annual interest rate (2.3% or 0.023 as a decimal),
- $ n $ is the number of compounding periods per year (52, for weekly compounding),
- $ t $ is the total time in years (15 years).

### Plugging in the values:

1. Convert the interest rate to decimal form:  
   $ r = 0.023 $
   
2. Substitute the values:
   - $ PMT = 120 $,
   - $ r = 0.023 $,
   - $ n = 52 $,
   - $ t = 15 $.

\[
FV = 120 \times \frac{(1 + \frac{0.023}{52})^{52 \times 15} - 1}{\frac{0.023}{52}}
\]

Now, calculating this expression step-by-step:

1. Calculate $ \frac{0.023}{52} $.
2. Calculate $ (1 + \frac{0.023}{52})^{780} $ (since $ 52 \times 15 = 780 $).
3. Subtract 1 from the result.
4. Divide by $ \frac{0.023}{52} $.
5. Multiply by 120.

Let's proceed with these calculations.The future value of the ordinary annuity, given the parameters, is **$111,745.44**.

Would you like further details on any step, or have any additional questions?

### Related Questions
1. How would the future value change if payments were made monthly instead of weekly?
2. What is the effect of increasing the interest rate to 3% on the future value?
3. How does the future value differ for an annuity due under the same conditions?
4. How much total interest is earned in this annuity over 15 years?
5. How would the outcome change if the time period were reduced to 10 years?

#### Tip:
When payments coincide with the compounding periods, the annuity grows more efficiently due to the compounding effect on each payment.

Find the future values of the ordinary annuities at the given annual rate r compounded as indicated. The payments are made to coincide with the periods of compounding. (Round your answer to the nearest cent.) PMT = $120, r = 2.3%, compounded weekly for 15 years

Learn to calculate the future value of an ordinary annuity with weekly compounding at an annual rate of 2.3% over 15 years, where payments of $120 coincide with compounding periods. This solution provides a step-by-step breakdown of the annuity formula, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation