You want to purchase a new car in 6 years and expect the car to cost ​$18 comma 000. Your bank offers a plan with a guaranteed APR of 5.5 % if you make regular monthly deposits. How much should you deposit each month to end up with ​$18 comma 000 in 6 ​years?
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Part 1
You should invest ​$
  
enter your response here each month.

To find out how much you need to deposit each month, we can use the formula for the future value of an ordinary annuity, which calculates the total amount accumulated by making regular monthly deposits:

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

Where:
- $FV$ is the future value (the amount you want to accumulate) = $18,000,
- $P$ is the monthly deposit amount (what we need to find),
- $r$ is the annual interest rate = 5.5% = 0.055,
- $n$ is the number of times the interest is compounded per year = 12 (monthly),
- $t$ is the number of years = 6.

Now, let's solve for $P$:

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

Let me calculate this for you.You should deposit approximately **$211.58** each month to accumulate $18,000 in 6 years with a 5.5% annual interest rate.

Do you need further details or explanations? Here are 5 related questions for expanding your understanding:

1. How does compound interest affect savings over time?
2. What would happen if the interest rate changed to 4%?
3. How is the future value formula different from the present value formula?
4. What are the effects of changing the deposit frequency (e.g., weekly instead of monthly)?
5. How much total interest will you earn over the 6-year period?

**Tip:** Regular small contributions can significantly grow over time due to compounding. Always look for higher APRs to maximize savings.

You want to purchase a new car in 6 years and expect the car to cost ​$18,000. Your bank offers a plan with a guaranteed APR of 5.5% if you make regular monthly deposits. How much should you deposit each month to end up with ​$18,000 in 6 years?

Calculate how much you need to deposit each month to save $18,000 in 6 years with a guaranteed 5.5% APR. This problem covers the future value of an annuity and compound interest, making it ideal for students in Grades 11-12 or college-level finance courses.

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