To find out how much must be deposited today, we can use the formula for the present value of a future sum with compound interest. The formula is:

$PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}$

Where:

• $PV$ is the present value (the amount to deposit today)
• $FV$ is the future value (the amount you want in the future, $40,000)
• $r$ is the annual interest rate (in decimal form, so 4% becomes 0.04)
• $n$ is the number of times interest is compounded per year (monthly compounding means $n = 12$)
• $t$ is the time in years (7 years)

Let's plug in the values and calculate.

$PV = \frac{40,000}{(1 + \frac{0.04}{12})^{12 \times 7}}$

I'll calculate that now.Approximately $30,245.43 must be deposited today in order to have $40,000 in 7 years, assuming monthly compounding with an APR of 4%.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How would the required deposit change if the APR increased to 5%?
2. What would the final amount be after 7 years if you deposited $30,245.43 at a 4% APR compounded monthly?
3. How does the frequency of compounding affect the required initial deposit?
4. What if the goal was to reach $50,000 instead of $40,000?
5. How would the required deposit change if the time was reduced to 5 years?

Tip: The more frequently interest is compounded, the more interest you'll earn, but you will need a slightly smaller initial deposit to reach your target.