Beginning in January, a person plans to deposit $1000 at the end of each month into an account earning 9% compounded monthly. Each year taxes must be paid on the interest earned during that year. Find the interest earned each year for the first 3 years.

To calculate the interest earned each year for the first three years, we need to break down the deposits, the compounded interest rate, and the effect of taxes. Here's a step-by-step approach to the calculation:

### Given:
- Monthly deposit: $ P = 1000 $
- Interest rate: $ r = 9\% $ per year (compounded monthly)
- Monthly interest rate: $ i = \frac{9\%}{12} = 0.75\% = 0.0075 $ per month
- Compounding frequency: 12 times per year
- Duration: 3 years
- We assume taxes are paid only on the interest earned at the end of each year, and the principal remains intact.

We can approach this problem using the formula for the future value of an annuity:

\[
FV = P \times \frac{(1 + i)^n - 1}{i}
\]

Where:
- $ FV $ is the future value after $ n $ months,
- $ P $ is the monthly payment,
- $ i $ is the monthly interest rate, and
- $ n $ is the number of months.

For each year, we will calculate:
1. The future value of the account at the end of the year.
2. Subtract the principal (total deposits) to determine the interest earned in that year.

#### **Step 1: Calculate for Year 1**
At the end of Year 1, 12 months have passed:
- $ n = 12 $
- Total deposit = $ 12 \times 1000 = 12,000 $

\[
FV_1 = 1000 \times \frac{(1 + 0.0075)^{12} - 1}{0.0075}
\]

Let me compute the result.At the end of Year 1, the future value of the account is approximately \$12,507.59.

Since the total deposits during Year 1 were \$12,000, the interest earned during Year 1 is:

\[
\text{Interest Year 1} = 12,507.59 - 12,000 = 507.59
\]

Thus, the interest earned in Year 1 is approximately **\$507.59**.

#### **Step 2: Calculate for Year 2**
At the end of Year 2, a total of 24 months have passed:
- $ n = 24 $
- Total deposits = $ 24 \times 1000 = 24,000 $

\[
FV_2 = 1000 \times \frac{(1 + 0.0075)^{24} - 1}{0.0075}
\]

Let me compute the result for Year 2.At the end of Year 2, the future value of the account is approximately \$26,188.47.

The total deposits by the end of Year 2 are \$24,000, so the interest earned during Year 2 is:

\[
\text{Interest Year 2} = 26,188.47 - 24,000 = 2,188.47
\]

Thus, the interest earned in Year 2 is approximately **\$2,188.47**.

#### **Step 3: Calculate for Year 3**
At the end of Year 3, a total of 36 months have passed:
- $ n = 36 $
- Total deposits = $ 36 \times 1000 = 36,000 $

\[
FV_3 = 1000 \times \frac{(1 + 0.0075)^{36} - 1}{0.0075}
\]

Let me compute the result for Year 3.At the end of Year 3, the future value of the account is approximately \$41,152.72.

The total deposits by the end of Year 3 are \$36,000, so the interest earned during Year 3 is:

\[
\text{Interest Year 3} = 41,152.72 - 36,000 = 5,152.72
\]

Thus, the interest earned in Year 3 is approximately **\$5,152.72**.

### Summary of Interest Earned:
1. Year 1: \$507.59
2. Year 2: \$2,188.47
3. Year 3: \$5,152.72

Would you like any additional details or explanations?

Here are some related questions you might be interested in:
1. How do taxes affect compounded interest?
2. How would the result change if the interest rate were compounded quarterly instead of monthly?
3. What is the future value if no withdrawals are made for 5 years?
4. How is the future value formula derived?
5. How does the frequency of compounding impact the total interest earned?

**Tip:** Compounding frequency has a significant impact on the growth of investments. Monthly compounding results in more frequent interest additions compared to yearly compounding.

Learn how to calculate the interest earned each year when depositing $1000 monthly at 9% compounded monthly. This detailed solution provides step-by-step calculations for interest earned in the first 3 years, covering core concepts like compound interest, the future value of an annuity, and monthly compounding. Suitable for advanced high school or college students.

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