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Let's solve the given problems step by step.

### Formula for compound interest:
1. **For discrete compounding**:
   \[
   A = P \left(1 + \frac{r}{n}\right)^{nt}
   \]
   Where:
   - $ A $ = the amount of money accumulated after $ t $ years, including interest.
   - $ P $ = the principal amount (initial investment).
   - $ r $ = annual interest rate (decimal form).
   - $ n $ = number of times interest is compounded per year.
   - $ t $ = time the money is invested for in years.

2. **For continuous compounding**:
   \[
   A = P e^{rt}
   \]
   Where:
   - $ e $ is Euler's number (approximately 2.71828).

### Given values:
- Principal ($ P $) = 2000 dollars.
- Interest rate ($ r $) = 4% = 0.04.
- Time ($ t $) = 9 years.

Now let's solve each case.

### 1. Compounded annually (n = 1):
Using the formula:
\[
A = 2000 \left(1 + \frac{0.04}{1}\right)^{1 \times 9}
\]

### 2. Compounded quarterly (n = 4):
\[
A = 2000 \left(1 + \frac{0.04}{4}\right)^{4 \times 9}
\]

### 3. Compounded monthly (n = 12):
\[
A = 2000 \left(1 + \frac{0.04}{12}\right)^{12 \times 9}
\]

### 4. Compounded continuously:
Using the formula for continuous compounding:
\[
A = 2000 e^{0.04 \times 9}
\]

Let me calculate these values.Here are the calculated amounts after 9 years with different compounding methods:

1. **Compounded annually**: \$2,846.62
2. **Compounded quarterly**: \$2,861.54
3. **Compounded monthly**: \$2,864.94
4. **Compounded continuously**: \$2,866.66

Would you like further details on any of these, or do you have additional questions?

### Related Questions:
1. What is the difference between annual and continuous compounding over longer periods?
2. How does the number of compounding periods affect the final amount?
3. Can you apply the same formulas for different interest rates?
4. What happens if interest is compounded daily?
5. How does changing the principal amount affect these calculations?

### Tip:
The more frequently interest is compounded, the greater the amount will be at the end of the investment period.

If 2000 dollars is invested in a bank account at an interest rate of 4 per cent per year, determine each of the following. Round each answer to the nearest cent.
1. Find the amount in the bank after 9 years if interest is compounded annually.
2. Find the amount in the bank after 9 years if interest is compounded quarterly.
3. Find the amount in the bank after 9 years if interest is compounded monthly.
4. Finally, find the amount in the bank after 9 years if interest is compounded continuously.

Learn how to calculate compound interest for $2000 invested over 9 years at a 4% annual interest rate, with detailed solutions for annual, quarterly, monthly, and continuous compounding. Explore key mathematical formulas and concepts such as exponential growth and interest compounding.

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