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To solve this problem, we need to calculate the future value of John's investment at the end of the given period, considering the different interest rates.

### Problem Recap:
- **Initial Deposit Date:** 01/07/2023
- **Monthly Deposit:** $500 at the beginning of each month
- **Interest Rates:**
  - $ j_1 = 2.91\% $ from 01/07/2023 to 09/30/2023
  - $ j_2 = 5.79\% $ from 10/01/2023 to 12/31/2023
- **Calculation Date:** 12/31/2023

We need to calculate the balance on 12/31/2023.

### Step 1: Calculate Future Value for the First Interest Period
For the first period (from 01/07/2023 to 09/30/2023), the interest rate $ j_1 $ is 2.91%. Since the deposits are made at the beginning of each month, we treat it as an annuity.

- **Interest Rate (monthly) for Period 1:** $ i_1 = \frac{2.91\%}{12} $
- **Number of months in Period 1:** 9 months

The future value (FV1) at the end of September 2023 can be calculated using the formula for the future value of an annuity:

\[
FV1 = P \times \frac{(1 + i_1)^n - 1}{i_1} \times (1 + i_1)
\]

### Step 2: Calculate Future Value for the Second Interest Period
After September 2023, the account will continue to earn interest at the new rate $ j_2 = 5.79\% $. We need to calculate the future value for the period from October to December (3 months).

- **Interest Rate (monthly) for Period 2:** $ i_2 = \frac{5.79\%}{12} $
- **Number of months in Period 2:** 3 months

We need to:
1. Compound the FV1 from September 2023 to December 2023.
2. Add the future value of the deposits made in October, November, and December.

### Step 3: Summing Up
Finally, add all the values to find the total balance on 12/31/2023. This balance should be rounded to three decimal places.

Let me perform these calculations now.The total balance amount in the account on 12/31/2023 will be approximately **$6135.690**.

Would you like further details or have any questions about this solution?

Here are some related questions you might find helpful:
1. How would the result change if the interest rates were compounded quarterly instead of monthly?
2. What would be the impact on the final amount if the monthly deposit was increased to $600?
3. How is the future value of an annuity calculated, and why is it different from simple compounding?
4. What happens if the interest rate changes every month instead of having fixed periods?
5. How would you calculate the balance if the deposits were made at the end of each month?
6. What is the difference between nominal and effective interest rates, and how does it affect calculations?
7. Can you explain the significance of compounding frequency in investment calculations?
8. How would the calculation differ if the interest rate was compounded continuously?

**Tip:** When dealing with financial calculations, always double-check the compounding periods and the timing of cash flows (beginning or end of the period). This ensures accuracy in predicting future values.

Learn how to calculate the future value of investments with variable interest rates. This detailed solution covers compound interest, annuities, and financial mathematics concepts. Suitable for advanced high school or college students.

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