The formula for the present value of money with compound interest is:

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

Where:

• $PV$ is the present value (the amount to invest now),
• $FV$ is the future value ($25,000 in this case),
• $r$ is the annual interest rate (7% or 0.07),
• $n$ is the number of compounding periods per year (quarterly means $n = 4$),
• $t$ is the number of years (10 years in this case).

Let's plug in the values and calculate:

$PV = \frac{25000}{(1 + \frac{0.07}{4})^{4 \times 10}}$

I'll calculate that for you.The amount you need to invest now in one lump sum to have $25,000 after 10 years with a 7% APR compounded quarterly is approximately $12,490.02.

Do you want any further details or explanations?

Here are 5 related questions:

1. How does the number of compounding periods affect the present value?
2. What would happen if the interest was compounded monthly instead of quarterly?
3. How would increasing the APR to 8% change the initial investment needed?
4. What is the future value if the investment is only $10,000?
5. How does the present value change if the investment period is 15 years instead of 10?

Tip: The more frequent the compounding, the more interest you will earn over time, which reduces the present value needed.