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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​$80 comma 000​, requals4.9​%, tequals3 yr
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Part 1
​a) The future value of the investment when interest is compounded annually is ​$
  
enter your response here.
​(Type an integer or a decimal. Round to the nearest cent as​ needed.)

To solve this problem, we use the formula for compound interest:

\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]

where:
- $ A $ is the future value of the investment
- $ P $ is the principal amount (initial investment)
- $ r $ is the annual interest rate (expressed as a decimal)
- $ n $ is the number of times the interest is compounded per year
- $ t $ is the time the money is invested for in years

Given:
- $ P = 80,000 $
- $ r = 4.9\% = 0.049 $
- $ t = 3 \, \text{years} $

Let's compute the future value for each part.

### Part (a) Annually Compounded Interest
For annual compounding, $ n = 1 $. Plugging the values into the formula:

\[
A = 80,000 \left( 1 + \frac{0.049}{1} \right)^{1 \times 3}
\]

I'll calculate this now.The future value of the investment when interest is compounded annually is approximately **$92,345.65**.

Would you like to proceed with part (b) for monthly compounding, or do you have any questions so far?

Here are 5 related questions to deepen your understanding:
1. How does the frequency of compounding affect the future value of an investment?
2. What is the difference between simple interest and compound interest?
3. How would the future value change if the investment period was extended to 5 years?
4. How does a higher interest rate impact the doubling time of an investment?
5. Why is continuous compounding often considered more advantageous in finance?

**Tip:** The more frequently interest is compounded, the higher the future value will be.

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. P = $80,000, r = 4.9%, t = 3 years.

Learn how to calculate the future value of an $80,000 investment with a 4.9% interest rate compounded annually, monthly, daily, and continuously over 3 years. Explore key formulas, including the compound interest formula and doubling time for investments.

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