Many persons prepare for retirement by making monthly contributions to a savings program. Suppose that 

​$2 comma 3002,300 

is set aside each year and invested in a savings account that pays 

2020​% 

interest per​ year, compounded continuously. **a.** The accumulated savings amount at the end of 

2323 

years will be 

​$10240001024000. 

​The 

value of the annuity will be

To calculate the value of the annuity, we can use the formula for the future value of continuous compounding interest in an annuity. The formula is:

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

Where:
- $ A $ is the accumulated value of the annuity.
- $ P $ is the annual contribution (in this case, $2,300).
- $ r $ is the interest rate (here, 20% or 0.20).
- $ t $ is the time in years (23 years).
- $ e $ is Euler's number (approximately 2.71828).

Given:
- $ P = 2,300 $
- $ r = 0.20 $
- $ t = 23 $
- The accumulated savings amount is $1,024,000.

Now, we can plug these values into the formula and calculate the result.

Let me calculate that for you.The accumulated value of the annuity at the end of 23 years is approximately **$1,132,569.63**.

Would you like more details on how this was calculated or have any questions? 

Here are 5 related questions for further exploration:
1. How does continuous compounding differ from regular compounding?
2. What if the interest rate changes to 15%? How would the accumulated value change?
3. How does the accumulated amount change if contributions are made monthly instead of annually?
4. What is the formula for discrete compounding in annuities, and how does it compare to continuous compounding?
5. How would early withdrawal penalties affect the annuity?

**Tip:** Continuous compounding results in the highest interest accumulation because it compounds at every possible moment.

Many persons prepare for retirement by making monthly contributions to a savings program. Suppose that $2,300 is set aside each year and invested in a savings account that pays 20% interest per year, compounded continuously. The accumulated savings amount at the end of 23 years will be $1,024,000. The value of the annuity will be?

Explore how to calculate the value of an annuity with continuous compounding at 20% interest over 23 years. Using the formula A = P * (e^(rt) - 1) / r, we compute the future value of $2,300 annual contributions and determine an accumulated savings of $1,024,000.

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