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Part 1
An initial investment amount​ P, an annual interest rate​ r, and a time t are given. Find the future value of the investment when interest is compounded​ (a) annually,​ (b) monthly,​ (c) daily, and​ (d) continuously. Then find​ (e) the doubling time T for the given interest rate.
Pequals​$4500​, requals4.15​%, tequals3 yr
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Part 1
​a) The future value of the investment when interest is compounded annually is ​$

enter your response here.
​(Type an integer or a decimal. Round to the nearest cent as​ needed.)

To find the future value of an investment with different compounding periods, we will use different formulas for each scenario. Let's start by defining the given values:

- Principal (initial investment) $ P = \$4500 $
- Annual interest rate $ r = 4.15\% = 0.0415 $
- Time $ t = 3 $ years

### (a) Compounded Annually

The formula for future value with annual compounding is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- $ n $ is the number of times interest is compounded per year (for annual compounding, $ n = 1 $)

Substituting the values, we get:
\[ A = 4500 \left(1 + \frac{0.0415}{1}\right)^{1 \cdot 3} \]
\[ A = 4500 \left(1 + 0.0415\right)^3 \]
\[ A = 4500 (1.0415)^3 \]

Now, we calculate $ (1.0415)^3 $:
\[ (1.0415)^3 \approx 1.129158 \]

So,
\[ A = 4500 \times 1.129158 \]
\[ A \approx 5081.21 \]

Therefore, the future value of the investment when interest is compounded annually is **\$5081.21**.

Would you like details or have any questions? Here are 5 questions for you to ask:
1. How do you calculate the future value with monthly compounding?
2. What is the formula for future value with daily compounding?
3. How do you determine the future value when interest is compounded continuously?
4. Can you explain how to find the doubling time for the given interest rate?
5. What are the key differences between different compounding intervals?

**Tip:** Always double-check your calculations to ensure accuracy, especially when working with exponents and percentages.

Learn how to calculate the future value of an investment with annual compounding using the compound interest formula. Given an initial investment amount of $4500, an annual interest rate of 4.15%, and a time period of 3 years, find the future value rounded to the nearest cent.

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