How much must be deposited at the beginning of each year in an account that pays 9%, compounded annually, so that the account will contain $28,000 at the end of 3 years? (Round your answer to the nearest cent.)

To solve this problem, we can use the **future value of an annuity due formula** since the deposits are made at the beginning of each period:

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

Where:
- $ FV $ = Future Value (the amount we want to achieve) = $28,000
- $ P $ = Deposit made at the beginning of each period (what we need to find)
- $ r $ = Interest rate per period = 9% or 0.09
- $ n $ = Number of periods = 3 years

Let's rearrange the formula to solve for $ P $:

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

### Substitute the values:

\[
P = \frac{28000}{\left(\frac{(1 + 0.09)^3 - 1}{0.09}\right) \times (1 + 0.09)}
\]

1. Calculate $ (1 + 0.09)^3 $:
   \[
   (1 + 0.09)^3 = 1.295029
   \]

2. Calculate $ (1.295029 - 1) / 0.09 $:
   \[
   \frac{1.295029 - 1}{0.09} = 3.2781
   \]

3. Multiply by $ (1 + 0.09) $:
   \[
   3.2781 \times 1.09 = 3.5731
   \]

4. Now, divide $ 28000 $ by $ 3.5731 $:
   \[
   P = \frac{28000}{3.5731} \approx 7835.14
   \]

### Answer:
The amount that must be deposited at the beginning of each year is approximately **$7,835.14**.

Would you like further details or have any questions?

Here are 5 additional questions to expand on this concept:
1. How would the deposit change if the interest rate increased to 10%?
2. What if the deposits were made at the end of each year instead of the beginning?
3. How much total interest is earned over the 3 years?
4. How would the calculation change if compounding occurred monthly instead of annually?
5. How much would need to be deposited each year if the goal was $50,000 instead of $28,000?

**Tip:** For annuity problems, remember that annuities due involve payments at the beginning of each period, while ordinary annuities involve payments at the end.

How much must be deposited at the beginning of each year in an account that pays 9%, compounded annually, so that the account will contain $28,000 at the end of 3 years? (Round your answer to the nearest cent.)

Find out how much you need to deposit at the beginning of each year in an account with 9% annual compounding to reach $28,000 in 3 years. This guide covers the future value of an annuity due formula, ideal for high school students learning about compound interest and time value of money.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation