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Part 1
Beginning in​ January, a person plans to deposit ​$100 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 during each year for the first 3 years.

To solve this problem, we'll first calculate the interest earned on monthly deposits over three years, considering the 9% annual interest rate compounded monthly. We'll then compute the total interest earned each year and determine the taxes owed.

### Step 1: Definitions
- Monthly deposit ($P$) = \$100
- Annual interest rate ($r$) = 9% = 0.09
- Compounding period per year = 12 (since interest is compounded monthly)
- Time in years ($n$) = 3 years

### Step 2: Calculate monthly interest rate
The monthly interest rate is the annual rate divided by 12:
\[
i = \frac{r}{12} = \frac{0.09}{12} = 0.0075
\]

### Step 3: Formula for the future value of an annuity
The future value ($FV$) of an annuity is given by the formula:
\[
FV = P \times \frac{(1 + i)^n - 1}{i}
\]
where:
- $P$ is the monthly deposit
- $i$ is the monthly interest rate
- $n$ is the total number of months

We will calculate the future value for each year and subtract the principal to determine the interest earned. 

Let me calculate the interest earned for each of the first three years.The interest earned during each of the first three years is as follows:

- **Year 1**: \$50.76
- **Year 2**: \$168.09
- **Year 3**: \$296.42

These values represent the interest accumulated for each year based on monthly deposits of \$100 into an account with 9% annual interest, compounded monthly.

Would you like more details, or have any questions?

Here are five related questions:
1. How is the compounding frequency affecting the interest calculations?
2. What would happen if the interest was compounded annually instead of monthly?
3. How does increasing the deposit amount change the interest earned?
4. Can we find the total amount in the account after 3 years, including both deposits and interest?
5. What is the impact of early withdrawals on the interest calculation?

**Tip**: The earlier you start depositing regularly, the more benefit you'll gain from compounding over time.

Beginning in January, a person plans to deposit $100 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 during each year for the first 3 years.

Learn how to calculate the interest earned each year for the first 3 years on monthly deposits of $100 into an account earning 9% annual interest, compounded monthly. Understand key financial concepts such as annuities and compound interest with step-by-step calculations. Suitable for advanced high school and college students.

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