Find the amount of an annuity if $200 per month is paid into it for a period of 17 years, earning interest at the rate of 7% per year compounded continuously. 

To find the amount of an annuity where regular payments are made and interest is compounded continuously, we use the formula for the future value of an annuity with continuous compounding. 

The formula is:

\[ A = P \left( \frac{e^{rt} - 1}{e^{\frac{r}{n}} - 1} \right) \]

where:
- $ A $ is the future value of the annuity,
- $ P $ is the payment amount per period,
- $ r $ is the annual interest rate (expressed as a decimal),
- $ t $ is the total number of years,
- $ n $ is the number of periods per year,
- $ e $ is the base of the natural logarithm.

Given:
- $ P = 200 $ dollars per month,
- $ r = 0.07 $ (7% annual interest rate),
- $ t = 17 $ years,
- $ n = 12 $ (monthly payments).

First, we need to convert the continuous compounding formula to fit the monthly payment scenario. For this, we use the adjusted formula for continuous compounding and monthly payments:

\[ A = P \cdot \frac{e^{rt} - 1}{r} \]

Substitute the given values into the formula:

\[ A = 200 \cdot \frac{e^{0.07 \cdot 17} - 1}{0.07} \]

First, calculate $ e^{0.07 \cdot 17} $:

\[ 0.07 \cdot 17 = 1.19 \]

\[ e^{1.19} \approx 3.286 \]

Next, calculate $ e^{1.19} - 1 $:

\[ 3.286 - 1 = 2.286 \]

Now, divide by the annual interest rate:

\[ \frac{2.286}{0.07} \approx 32.657 \]

Finally, multiply by the monthly payment amount $ P $:

\[ A = 200 \cdot 32.657 \approx 6531.40 \]

This gives the future value of the annuity with monthly payments:

\[ A \approx 6531.40 \]

However, this result looks suspicious because we should also account for the number of periods in the time span of 17 years, implying a more careful approach:

Let's re-calculate using the continuous compounding formula which directly applies to monthly deposits over continuous compounding:

\[ A = P \left( \frac{e^{rt} - 1}{e^{r/12} - 1} \right) \]

\[ A = 200 \left( \frac{e^{0.07 \times 17} - 1}{e^{0.07/12} - 1} \right) \]

Calculate each step precisely:

\[ e^{0.07/12} = e^{0.0058333} \approx 1.005846 \]

\[ e^{1.19} \approx 3.286 \]

Thus:

\[ \frac{e^{0.07 \times 17} - 1}{e^{0.0058333} - 1} = \frac{3.286 - 1}{1.005846 - 1} = \frac{2.286}{0.005846} \approx 391.15 \]

So,

\[ A = 200 \times 391.15 \approx 78230 \]

Therefore, the future value of the annuity is approximately:

\[ A \approx \$78,230 \]

This correctly reflects the amount of an annuity with continuous compounding over the years.

Do you have any questions or need further details on any part of the solution?

Here are 8 related questions you might find interesting:
1. What would the future value be if the interest rate were 6% instead of 7%?
2. How does the compounding frequency affect the future value of the annuity?
3. What is the formula for the future value of an annuity with discrete compounding?
4. How would the future value change if the payments were made annually instead of monthly?
5. What is the present value of the annuity given the same parameters?
6. How does inflation impact the real value of the annuity's future value?
7. How would the future value change if the annuity were only for 10 years instead of 17 years?
8. What would happen to the future value if an additional $50 is added to each monthly payment?

**Tip:** Always ensure to convert interest rates to match the compounding period and payment period for accurate calculations.

Learn how to calculate the future value of an annuity with continuous compounding using a detailed step-by-step solution. This problem involves monthly payments, an annual interest rate, and continuous compounding. Suitable for advanced high school students or college-level learners.

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