Today is 1/7/2023, John plans to deposit $500 at the beginning of each month into an investment fund. The first deposit will be deposited today. John predicts that the return rate of this fund will be j2=3.47% from 1/7/2023 to 30/9/2023 and  j2=3.04% from 1/10/2023 to 31/12/2023. What will be the balance amount of this account on 31/12/2023? Round your answer to three decimal places. 


Let's calculate the balance of John's account on 31/12/2023, considering the monthly deposits and the given interest rates.

### Given Information:
- **Monthly Deposit:** $500 (beginning of each month)
- **First Deposit:** 1/7/2023
- **First Interest Rate (1/7/2023 - 30/9/2023):** j2 = 3.47%
- **Second Interest Rate (1/10/2023 - 31/12/2023):** j2 = 3.04%

### Step 1: Convert the nominal annual interest rates (j2) to monthly effective rates.

For a nominal semi-annual interest rate $ j2 $, the monthly effective interest rate $ i_{\text{monthly}} $ is calculated as:

\[
i_{\text{monthly}} = \left(1 + \frac{j2}{2}\right)^{\frac{1}{6}} - 1
\]

For the first period (July to September):
\[
i_{\text{monthly,1}} = \left(1 + \frac{0.0347}{2}\right)^{\frac{1}{6}} - 1 \approx 0.002869
\]

For the second period (October to December):
\[
i_{\text{monthly,2}} = \left(1 + \frac{0.0304}{2}\right)^{\frac{1}{6}} - 1 \approx 0.002512
\]

### Step 2: Calculate the future value at the end of each period.

#### Future Value at the End of September (First Period):

\[
FV_{\text{Sep}} = 500 \times (1 + 0.002869)^2 + 500 \times (1 + 0.002869) + 500
\]

Calculating:

\[
FV_{\text{Sep}} = 500 \times 1.00575 + 500 \times 1.002869 + 500 \approx 502.875 + 501.435 + 500 \approx 1504.31
\]

#### Future Value at the End of December (Second Period):

Now, we'll use the balance from September to calculate the balance at the end of December with the new interest rate:

\[
FV_{\text{Dec}} = \left(1504.31 + 500 \times (1 + 0.002512)^2 + 500 \times (1 + 0.002512) + 500\right) \times (1 + 0.002512)
\]

Calculating:

\[
FV_{\text{Dec}} \approx \left(1504.31 + 502.515 + 501.256 + 500\right) \times 1.002512 \approx 3007.50 \times 1.002512 \approx 3027.096
\]

### Final Answer:

The balance amount of the account on 31/12/2023 is approximately **$3027.096**.

---

Do you need more details or have any questions?

Here are some related questions you might find helpful:

1. How does the interest rate affect the final balance in an investment account?
2. What is the difference between nominal and effective interest rates?
3. How would the balance change if John deposited at the end of each month instead of the beginning?
4. What would the balance be if John only deposited for six months?
5. How can John calculate the compound interest for different time intervals?
6. What if the interest rate changes more frequently within the investment period?
7. How would different deposit amounts affect the future value?
8. What are the benefits of making regular investments?

**Tip:** Always keep track of when interest rates change, as it significantly impacts the growth of your investments.

Learn how to calculate the balance of an investment account with monthly deposits and varying interest rates. Follow a detailed example from July 1, 2023, to December 31, 2023, with a step-by-step solution. Suitable for advanced high school students studying finance or mathematics.

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