This problem involves calculating the future value of a series of cash flows with interest compounded monthly. There are two components to this investment:

1. Initial deposit of $7000: This deposit will accumulate interest over the full 4 years.
2. Monthly deposits of $225: These are made at the end of each month for 4 years.

The formula for the future value $FV$ of a compound interest investment is:

$FV = P \times \left(1 + \frac{r}{n}\right)^{nt}$

Where:

• $P$ is the initial principal (in this case, $7000),
• $r$ is the annual interest rate (3% or 0.03),
• $n$ is the number of times the interest is compounded per year (monthly, so $n = 12$),
• $t$ is the time the money is invested or borrowed for (in years).

The second part involves calculating the future value of an ordinary annuity (monthly deposits of $225). The formula for the future value of an annuity is:

$FV_{\text{annuity}} = PMT \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}$

Where:

• $PMT$ is the monthly payment ($225),
• $r$ is the annual interest rate (0.03),
• $n$ is the number of compounding periods per year (12),
• $t$ is the number of years (4).

Now let's calculate both parts.

Step 1: Calculate Future Value of the Initial Deposit

Using the compound interest formula:

$FV_{\text{initial}} = 7000 \times \left(1 + \frac{0.03}{12}\right)^{12 \times 4}$

Step 2: Calculate Future Value of the Monthly Deposits

Using the future value of an annuity formula:

$FV_{\text{annuity}} = 225 \times \frac{\left(1 + \frac{0.03}{12}\right)^{12 \times 4} - 1}{\frac{0.03}{12}}$

Step 3: Add the Results Together

The total future value is the sum of the future values from both the initial deposit and the monthly deposits.

Let's calculate.Lauren will have approximately $19,350.82 in her account at the end of 4 years.

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

Here are 5 related questions to expand this information:

1. How would the result change if Lauren made her monthly deposits at the beginning of each month instead of the end?
2. What if the interest rate was 4% instead of 3%?
3. How does compounding frequency (monthly vs. quarterly) affect the future value?
4. How much of the final amount is from interest earned?
5. What happens if Lauren skips some of the monthly payments?

Tip: Increasing the frequency of compounding generally increases the future value due to interest being applied more often.