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?

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.