Find the future value of each annuity due. Then determine how much of this value is from contributions and how much is from interest.
Payments of ​$240 made at the beginning of each quarter for 12 years at 3.4​% compounded quarterly
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Part 1
The future value of the annuity due is ​$
  
enter your response here.
​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

To solve this, let's start with the formula for the future value of an annuity due:

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

where:
- $ P $ is the payment amount per period,
- $ i $ is the interest rate per period,
- $ n $ is the total number of payments,
- $ FV $ is the future value of the annuity due.

Given:
- $ P = 240 $,
- Interest rate per year = $ 3.4\% $,
- Compounded quarterly, so $ i = \frac{3.4\%}{4} = 0.85\% = 0.0085 $,
- Periods per year = 4,
- Total time = 12 years, so $ n = 12 \times 4 = 48 $ quarters.

### Step 1: Substitute values into the formula
\[
FV = 240 \times \frac{(1 + 0.0085)^{48} - 1}{0.0085} \times (1 + 0.0085)
\]

### Step 2: Calculate the future value

Let's go through the calculations:

1. Calculate $ (1 + 0.0085)^{48} $.
2. Subtract 1 from this result.
3. Divide by 0.0085.
4. Multiply by 240.
5. Finally, multiply by $ (1 + 0.0085) $.

Let me calculate this.The future value of the annuity due is approximately **$14,272.29**.

### Part 2: Breakdown of Contributions and Interest

To determine how much of this future value comes from contributions and how much is from interest, we need to calculate:
1. **Total contributions**: This is simply the payment amount $ P $ multiplied by the number of payments $ n $.
2. **Interest earned**: Subtract the total contributions from the future value.

#### Step-by-Step Calculation

1. **Total Contributions**: 
   \[
   \text{Total Contributions} = P \times n = 240 \times 48
   \]

2. **Interest Earned**:
   \[
   \text{Interest Earned} = \text{Future Value} - \text{Total Contributions}
   \]

Let's calculate these values.1. **Total Contributions**: $11,520
2. **Interest Earned**: Approximately $2,752.29

### Summary:
- The **future value** of the annuity due is **$14,272.29**.
- **Total Contributions**: $11,520
- **Interest Earned**: $2,752.29

Would you like further details on any part of this calculation?

### Related Questions
1. How would the future value change if the interest rate were higher, say 5%?
2. How does changing the compounding period (e.g., monthly instead of quarterly) affect the future value?
3. What is the difference in future value between an annuity due and an ordinary annuity?
4. How would an increase in the payment amount affect the future value?
5. What is the formula to calculate the present value of an annuity due?

#### Tip:
When dealing with annuities, always verify whether it is an annuity due or an ordinary annuity, as it affects the timing and thus the future value calculation.

Find the future value of each annuity due. Then determine how much of this value is from contributions and how much is from interest. Payments of $240 made at the beginning of each quarter for 12 years at 3.4% compounded quarterly.

Discover how to calculate the future value of an annuity due with $240 payments made at the beginning of each quarter for 12 years at an interest rate of 3.4%. This guide breaks down the total contributions and interest earned.

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