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Let's work through the problem step by step.

The formula for compound interest is:

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

Where:
- \( A \) = final amount
- \( P \) = initial principal (2000 dollars)
- \( r \) = annual interest rate (4% = 0.04)
- \( n \) = number of times the interest is compounded per year
- \( t \) = number of years (9 years)

### 1. **Compounded Annually**:
Here, \( n = 1 \).
\[
A = 2000 \left(1 + \frac{0.04}{1}\right)^{1 \times 9}
\]
\[
A = 2000 \left(1.04\right)^9
\]
Calculating this gives us the amount.

### 2. **Compounded Quarterly**:
Here, \( n = 4 \).
\[
A = 2000 \left(1 + \frac{0.04}{4}\right)^{4 \times 9}
\]
\[
A = 2000 \left(1 + 0.01\right)^{36}
\]
Calculating this gives the quarterly-compounded amount.

### 3. **Compounded Monthly**:
Here, \( n = 12 \).
\[
A = 2000 \left(1 + \frac{0.04}{12}\right)^{12 \times 9}
\]
\[
A = 2000 \left(1 + 0.00333\right)^{108}
\]
Calculating this gives the monthly-compounded amount.

### 4. **Compounded Continuously**:
For continuous compounding, the formula is:
\[
A = P e^{rt}
\]
Where:
- \( e \) is the mathematical constant (approximately 2.71828).

\[
A = 2000 \times e^{0.04 \times 9}
\]
Calculating this gives the continuously compounded amount.

Would you like me to calculate each of these for you now?

Here are 5 relative questions:
1. How does the number of compounding periods affect the final amount?
2. What is the impact of a higher interest rate on the compounded interest?
3. How would the results change if the investment period were 15 years instead of 9 years?
4. What happens if the interest rate is compounded daily (365 times a year)?
5. How does continuous compounding differ fundamentally from regular compounding?

**Tip**: Compounding more frequently increases the total amount earned, though the difference becomes smaller as the frequency increases.

If 2000 dollars is invested in a bank account at an interest rate of 4 per cent per year, find the amount in the bank after 9 years if interest is compounded annually, quarterly, monthly, and continuously.

Learn how to calculate compound interest for an investment of 2000 dollars at 4% per year over 9 years, with annual, quarterly, monthly, and continuous compounding. Understand key concepts in compound interest and exponential growth, suitable for high school math students.

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