Beginning in​ January, a person plans to deposit 

​$10 

at the end of each month into an account earning 

6​% 

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. find the interesrt earned during the second year 

To find the interest earned during the second year, we need to calculate how the account balance grows with monthly deposits and monthly compounding interest. Since this is a compound interest problem with regular deposits, we use the **future value of an annuity formula** to track the growth of the account.

Here’s a step-by-step outline for solving this problem:

### 1. Monthly Deposit and Interest Rate
- Monthly deposit ($D$) = $10
- Annual interest rate ($r$) = 6%, so the **monthly interest rate** is:
  \[
  i = \frac{6\%}{12} = 0.5\% = 0.005
  \]

### 2. Balance Growth Formula (Future Value of Annuity)
The future value of an annuity formula for monthly deposits is:
\[
FV = D \times \frac{(1 + i)^n - 1}{i}
\]
Where:
- $FV$ = future value of the annuity after $n$ months
- $D$ = monthly deposit ($10)
- $i$ = monthly interest rate (0.005)
- $n$ = number of months

### 3. Interest Earned During Year 2
To find the interest earned during the second year, we'll calculate the account balance at the end of Year 1 and Year 2, and then find the difference due to interest alone.

#### End of Year 1 (after 12 months):
\[
FV_{1} = 10 \times \frac{(1 + 0.005)^{12} - 1}{0.005}
\]

#### End of Year 2 (after 24 months):
\[
FV_{2} = 10 \times \frac{(1 + 0.005)^{24} - 1}{0.005}
\]

### 4. Interest Earned in Year 2
The interest earned in Year 2 will be the difference in the balance between the end of Year 2 and the amount contributed during the second year (12 deposits of $10), minus the balance at the end of Year 1.

Let me calculate this for you.The calculations give us the following:

- The balance at the end of **Year 1** is approximately **$123.36**.
- The balance at the end of **Year 2** is approximately **$254.32**.
- The interest earned during **Year 2** is approximately **$10.96**.

Would you like more details on the calculations or have any other questions?

Here are 5 related questions to expand on the topic:
1. How is compound interest different from simple interest?
2. What would be the total interest earned over 3 years?
3. How does the frequency of compounding affect the final amount in an account?
4. How can tax rates on interest impact overall earnings?
5. What is the future value after 5 years with the same deposits and interest rate?

**Tip:** To maximize interest, consider making more frequent deposits or finding an account with a higher interest rate.

Beginning in January, a person plans to deposit $10 at the end of each month into an account earning 6% 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. Find the interest earned during the second year.

Learn how to calculate the interest earned on monthly deposits of $10 in an account with 6% annual interest compounded monthly. This problem demonstrates the use of the future value of annuity formula and explores the interest earned over the first three years, with a focus on Year 2.

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