Beginning in​ January, a person plans to deposit 

​$100100 

at the end of each month into an account earning 

15​% 

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 need to calculate the interest earned each year for the first three years based on monthly deposits of \$100,100 into an account with 15% interest compounded monthly.

### Step-by-step solution:

#### **1. Monthly Interest Rate**
The annual interest rate is 15%, so the monthly interest rate will be:
\[
r = \frac{15\%}{12} = \frac{0.15}{12} = 0.0125
\]

#### **2. Formula for Compound Interest with Monthly Deposits**
The amount in the account after each month can be calculated using the formula for future value with regular monthly deposits, compounded monthly:
\[
A_n = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- $A_n$ = the amount after $n$ months
- $P$ = the monthly deposit (\$100,100)
- $r$ = the monthly interest rate (0.0125)
- $n$ = the number of months

#### **3. Interest Calculation Each Year**

Let’s compute the interest earned in each year by finding the total amount at the end of each year and then subtracting the total deposits made.

##### **Year 1:**
In the first year, there are 12 months of deposits. The total deposits at the end of 12 months are:
\[
\text{Total deposits} = 12 \times 100100 = 1201200
\]
Using the future value formula for monthly compounded interest:
\[
A_{12} = 100100 \times \frac{(1 + 0.0125)^{12} - 1}{0.0125}
\]

Let’s calculate the exact value for $A_{12}$.At the end of **Year 1**, the total amount in the account is approximately \$1,287,322.18. Since the total deposits were \$1,201,200, the interest earned during Year 1 is approximately **\$86,122.18**.

##### **Year 2:**
Now, for Year 2, we have 24 months in total, so we need to calculate the amount after 24 months and subtract the total deposits made by the end of Year 2:
\[
\text{Total deposits by Year 2} = 24 \times 100100 = 2402400
\]
Using the future value formula again:
\[
A_{24} = 100100 \times \frac{(1 + 0.0125)^{24} - 1}{0.0125}
\]

Let’s compute the amount and interest earned by the end of Year 2.At the end of **Year 2**, the total amount in the account is approximately \$2,781,587.21. Since the total deposits by the end of Year 2 were \$2,402,400, the interest earned during Year 2 is approximately **\$379,187.21**.

##### **Year 3:**
For Year 3, we consider 36 months in total. The total deposits by the end of Year 3 are:
\[
\text{Total deposits by Year 3} = 36 \times 100100 = 3603600
\]
Again, using the future value formula:
\[
A_{36} = 100100 \times \frac{(1 + 0.0125)^{36} - 1}{0.0125}
\]

Let’s compute the amount and interest earned by the end of Year 3.At the end of **Year 3**, the total amount in the account is approximately \$4,516,062.10. Since the total deposits by the end of Year 3 were \$3,603,600, the interest earned during Year 3 is approximately **\$912,462.10**.

### Summary of Interest Earned:
1. **Year 1:** \$86,122.18
2. **Year 2:** \$379,187.21
3. **Year 3:** \$912,462.10

Would you like more details on any of the calculations?

Here are five related questions to consider:
1. How does the interest calculation change with a different interest rate?
2. What if the deposits are made at the beginning of the month instead of the end?
3. How does the total interest compare over 10 years with the same monthly deposit?
4. What if the interest were compounded daily instead of monthly?
5. How would additional taxes affect the final interest over three years?

**Tip:** The earlier the deposits, the more compounding periods you have, resulting in more interest earned.

Beginning in January, a person plans to deposit $100,100 at the end of each month into an account earning 15% 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.

Calculate the interest earned on monthly deposits of $100,100 with a 15% annual interest rate compounded monthly. Discover how much interest is earned each year for the first 3 years, using detailed calculations based on compound interest formulas. Suitable for students in Grades 10-12.

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