https://file.aiquickdraw.com/mathbot/file/1730484555455w650te8p.jpg

To solve this annuity problem, let's use the annuity future value formula shown in the image:

\[
A = \text{Pay} \times \left( \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}} \right)
\]

where:
- $ A $ is the final amount.
- $\text{Pay}$ is the regular deposit (payment) amount.
- $ r $ is the annual interest rate (in decimal form).
- $ n $ is the number of compounding periods per year.
- $ t $ is the number of years.

### Given Values
For this problem:
- $\text{Pay} = 800$ dollars (deposited quarterly).
- $ r = 5.6\% = 0.056$ (annual interest rate in decimal form).
- $ n = 4$ (since interest is compounded quarterly).
- $ t = 18$ years.

### Step-by-Step Calculation
1. **Convert the Interest Rate per Compounding Period:**
   \[
   \frac{r}{n} = \frac{0.056}{4} = 0.014
   \]

2. **Calculate the Total Number of Compounding Periods:**
   \[
   nt = 4 \times 18 = 72
   \]

3. **Apply the Values to the Formula:**
   Substitute these values into the formula:
   \[
   A = 800 \times \left( \frac{\left(1 + 0.014\right)^{72} - 1}{0.014} \right)
   \]

4. **Compute the Inner Exponent:**
   \[
   1 + 0.014 = 1.014
   \]
   \[
   1.014^{72} \approx 2.8966
   \]

5. **Calculate the Formula's Components:**
   \[
   2.8966 - 1 = 1.8966
   \]
   \[
   \frac{1.8966}{0.014} \approx 135.4714
   \]

6. **Calculate the Final Amount:**
   \[
   A = 800 \times 135.4714 \approx 108,377.12
   \]

### Answer
The final amount $ A $ after 18 years is approximately **$108,377.12**.

Would you like a breakdown of any specific step? 

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Here are five questions that extend this concept:

1. How would the final amount change if the compounding frequency increased to monthly?
2. What is the impact of increasing the interest rate by 1%?
3. How would the result differ if the deposit was made semi-annually instead of quarterly?
4. What would the accumulated amount be after only 10 years under the same conditions?
5. How would changing the deposit amount to $1,000 affect the final balance?

**Tip:** In annuity calculations, increasing the compounding frequency (e.g., quarterly to monthly) generally leads to a higher final amount due to more frequent application of interest.

$800 deposited quarterly earning 5.6% compounded quarterly for 18 years.

Learn how to calculate the future value of an annuity with $800 deposited quarterly at 5.6% interest compounded quarterly over 18 years. This problem demonstrates the use of the annuity future value formula and covers key concepts like compound interest and exponential growth. Suitable for high school students in Grades 11-12.

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