Sue will need $\mathbf{\$ 1 , 0 0 0}$ for down payment on a new car in $\mathbf{3}$ years. What monthly deposi should she make in an account that pays $6 \%$ compounded monthly so that she receives the needed money after 3 years? Each payment is made at the beginning of each period (Round the answer upto 2 decimal places.)
Done

Step 1 ：We are given that Sue needs to save $1000 for a down payment on a new car in 3 years. She plans to make monthly deposits into an account that pays 6% interest compounded monthly. We are asked to find out how much she should deposit each month.

Step 2 ：We can use the formula for the future value of a series of payments (or annuity) to solve this problem. The formula is: \(FV = P \times \frac{(1 + r/n)^{n \times t} - 1}{r/n}\), where FV is the future value, P is the monthly deposit, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.

Step 3 ：We can rearrange this formula to solve for P: \(P = FV \times \frac{r/n}{(1 + r/n)^{n \times t} - 1}\)

Step 4 ：Substituting the given values into the formula, we get: \(P = 1000 \times \frac{0.06/12}{(1 + 0.06/12)^{12 \times 3} - 1}\)

Step 5 ：Solving this equation gives us the monthly deposit that Sue should make. After rounding to two decimal places, we find that Sue should make a monthly deposit of approximately \$25.42.

Step 6 ：Final Answer: Sue should make a monthly deposit of approximately \(\boxed{25.42}\) to have \$1000 after 3 years.