Lauren plans to deposit $6000 into a bank account at the beginning of next month and $250/month into the same account at the end of that month and at the end of each subsequent month for the next 3 years. If her bank pays interest at a rate of 4%/year compounded monthly, how much will Lauren have in her account at the end of 3 years? (Assume she makes no withdrawals during the 3-year period. Round your answer to the nearest cent.)

To determine how much Lauren will have in her account at the end of 3 years, we need to calculate the future value of two different cash flows:

1. **The lump-sum deposit of $6,000** at the beginning of the period.
2. **The series of monthly deposits of $250** made at the end of each month.

### Step 1: Future Value of the Lump-Sum Deposit
The formula for the future value of a lump sum is:

\[
FV = PV \times \left( 1 + \frac{r}{n} \right)^{nt}
\]

Where:
- $ PV = 6000 $ (initial deposit)
- $ r = 0.04 $ (annual interest rate)
- $ n = 12 $ (number of times interest is compounded per year, i.e., monthly)
- $ t = 3 $ (number of years)

Now let's calculate the future value of this lump sum.

\[
FV_1 = 6000 \times \left(1 + \frac{0.04}{12}\right)^{12 \times 3}
\]

### Step 2: Future Value of the Monthly Deposits
The formula for the future value of a series of payments (ordinary annuity) is:

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

Where:
- $ P = 250 $ (monthly deposit)
- $ r = 0.04 $ (annual interest rate)
- $ n = 12 $ (number of times interest is compounded per year)
- $ t = 3 $ (number of years)

Now let's calculate the future value of the monthly deposits.

\[
FV_2 = 250 \times \frac{\left(1 + \frac{0.04}{12}\right)^{12 \times 3} - 1}{\frac{0.04}{12}}
\]

Finally, the total future value is the sum of $ FV_1 $ and $ FV_2 $.

Let’s calculate the total amount.At the end of 3 years, Lauren will have approximately **$16,309.02** in her bank account.

Would you like more details on the calculation? 

Here are 5 related questions:
1. How does the compounding frequency affect the total future value of an investment?
2. What is the impact of increasing monthly deposits on the future value of the account?
3. What would happen if the interest rate were compounded quarterly instead of monthly?
4. How would the future value change if Lauren extended her deposits for an additional 2 years?
5. What is the future value of the account if Lauren starts with a higher initial deposit?

**Tip**: Regular monthly deposits into a compound interest account can greatly enhance your savings over time due to the effects of interest compounding.

Discover how to calculate the future value of an initial lump sum and monthly deposits into a savings account with 4% annual interest compounded monthly. Includes detailed steps and formulas for finding the future value of both a lump sum and an annuity. Suitable for Grades 10-12.

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