Your goal is to create a college fund for your child. Suppose you find a fund that offers an APR of 6 %. How much should you deposit monthly to accumulate ​$82 comma 000 in 14 ​years?
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Part 1
You should invest ​$

enter your response here each month.
​(Do not round until the final answer. Then round to two decimal places as​ needed.)

To calculate how much you should deposit monthly to accumulate $82,000 in 14 years with an annual percentage rate (APR) of 6%, we will use the formula for the future value of a series of regular monthly deposits into an account earning compound interest.

The formula is:

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

Where:
- $ FV $ is the future value (in this case, $82,000),
- $ P $ is the monthly deposit (which we are solving for),
- $ r $ is the annual interest rate (6%, or 0.06),
- $ n $ is the number of times interest is compounded per year (monthly, so 12),
- $ t $ is the number of years (14 years).

Let's solve for $ P $.

\[
82,000 = P \times \frac{(1 + 0.06/12)^{12 \times 14} - 1}{0.06/12}
\]

Now, let's calculate it.You should invest approximately **$312.61** each month to accumulate $82,000 in 14 years with a 6% annual interest rate.

Would you like further details or have any questions? 

Here are 5 related questions for exploration:
1. How would the required monthly deposit change if the interest rate were 5% instead of 6%?
2. What would happen to the total savings if you deposited an additional $50 each month?
3. How would a shorter investment period, such as 10 years, affect the monthly deposit?
4. How much interest will you have earned after the 14 years?
5. How does the formula change if the interest compounds quarterly instead of monthly?

**Tip:** Regularly reviewing the interest rate and making adjustments to the monthly deposit can help you stay on track toward your savings goal.

Your goal is to create a college fund for your child. Suppose you find a fund that offers an APR of 6 %. How much should you deposit monthly to accumulate $82,000 in 14 years?

Learn how to calculate the monthly deposits required to accumulate $82,000 in 14 years with an APR of 6%. This detailed solution uses the future value of annuity formula and demonstrates the power of compound interest.

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