What is the accumulated value of periodic deposits of $30 at the beginning of every quarter for 17 years if the interest rate is 4.03% compounded quarterly?

To find the accumulated value of periodic deposits at the beginning of every quarter, we can use the future value of an annuity due formula. The formula is:

\[
FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r)
\]

Where:
- $ FV $ is the future value of the annuity due.
- $ P $ is the periodic deposit.
- $ r $ is the interest rate per period.
- $ n $ is the total number of deposits.

### Given:
- $ P = 30 $ (the amount of each deposit)
- Annual interest rate = 4.03%, so the quarterly interest rate $ r = \frac{4.03\%}{4} = 0.010075 $
- Total number of years = 17, so total number of quarters $ n = 17 \times 4 = 68 $

### Plugging in the values:
1. Calculate $ r $:
   \[
   r = \frac{4.03}{100} \div 4 = 0.010075
   \]

2. Calculate $ n $:
   \[
   n = 17 \times 4 = 68
   \]

3. Calculate $ FV $:
   \[
   FV = 30 \times \left( \frac{(1 + 0.010075)^{68} - 1}{0.010075} \right) \times (1 + 0.010075)
   \]

Calculating $ (1 + 0.010075)^{68} $:
\[
(1.010075)^{68} \approx 1.8983
\]

Now substituting:
\[
FV = 30 \times \left( \frac{1.8983 - 1}{0.010075} \right) \times 1.010075
\]
\[
= 30 \times \left( \frac{0.8983}{0.010075} \right) \times 1.010075
\]
\[
= 30 \times 89.1 \times 1.010075 \approx 30 \times 89.93 \approx 2697.90
\]

Thus, the accumulated value of the periodic deposits is approximately **$2,697.90**.

### Would you like to explore more about this calculation or related concepts? Here are some questions you might consider:
1. How does changing the interest rate affect the accumulated value?
2. What would be the value if the deposits were made at the end of each period instead of the beginning?
3. How does the frequency of deposits influence the total amount accumulated?
4. Can you explain the difference between simple interest and compound interest?
5. What are other investment strategies that could yield a similar or higher return?

**Tip:** Regularly reviewing your investment strategy and the interest rates can help optimize your savings over time.

Explore how to calculate the accumulated value of periodic deposits of $30 every quarter for 17 years at an interest rate of 4.03% compounded quarterly. This comprehensive guide covers annuities and compound interest formulas, suitable 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