https://file.aiquickdraw.com/mathbot/file/1722366078151qjxhpls4.png

To solve this problem, we need to use the formula for **continuous compounding interest**. This formula is used to calculate the present value (amount that needs to be invested now) when the interest is compounded continuously. The formula is:

\[
A = Pe^{rt}
\]

where:
- $ A $ is the future value (amount desired in the future),
- $ P $ is the principal (amount invested now),
- $ r $ is the annual interest rate (expressed as a decimal),
- $ t $ is the time in years,
- $ e $ is the base of the natural logarithm (approximately 2.71828).

### Given Data:
- Future Value, $ A = 2500 $
- Annual Interest Rate, $ r = 6.5\% = 0.065 $
- Time, $ t = 2 $ years

We need to find the principal $ P $.

### Step 1: Set up the equation

Rearrange the formula to solve for $ P $:

\[
P = \frac{A}{e^{rt}}
\]

### Step 2: Substitute the given values

\[
P = \frac{2500}{e^{0.065 \times 2}}
\]

### Step 3: Calculate the exponent

\[
e^{0.065 \times 2} = e^{0.13}
\]

Calculate $ e^{0.13} $:

\[
e^{0.13} \approx 1.13883
\]

### Step 4: Solve for $ P $

\[
P = \frac{2500}{1.13883}
\]

\[
P \approx 2194.33
\]

### Conclusion

Therefore, you should invest approximately **$2194.33** now at an interest rate of 6.5% per year, compounded continuously, to have $2500 in two years.

**Final Answer: $2194.33**

### Would you like more details or have any questions?

### Related Questions:

1. What is the difference between continuous compounding and other types of compounding?
2. How does changing the interest rate affect the required principal amount?
3. What is the formula for compounding interest compounded annually?
4. Can you calculate the amount after 5 years if you invest $3000 at 6.5% compounded continuously?
5. How can you find the time required to reach a certain amount with continuous compounding?
6. What happens if the interest rate changes to 7%? How much should be invested?
7. What is the continuous compounding formula used for?
8. How does continuous compounding compare to daily compounding?

### Tip:

When dealing with problems involving continuous compounding, remember that the constant $ e $ plays a crucial role. It represents an exponential growth scenario where compounding happens at every possible moment.

Learn how to calculate continuous compounding interest with a step-by-step solution. Given a future value of $2500, an annual interest rate of 6.5%, and a time of 2 years, find out how much to invest now. Suitable for advanced high school students and beyond.

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