The present value (PV) formula for compound interest is given by:

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

where:

• $PV$ is the present value (the amount to invest now),
• $FV$ is the future value (the amount you want in the future, which is $25,000 in this case),
• $r$ is the annual percentage rate (APR), given as 9% or 0.09,
• $n$ is the number of compounding periods per year (since it's compounded quarterly, $n = 4$),
• $t$ is the number of years (10 years here).

Step-by-step calculation:

1. Identify variables:

   • $FV = 25,000$
   • $r = 0.09$
   • $n = 4$
   • $t = 10$

2. Plug into the formula:

$PV = \frac{25,000}{(1 + \frac{0.09}{4})^{4 \times 10}}$

3. Simplify and calculate:

$PV = \frac{25,000}{(1 + 0.0225)^{40}} = \frac{25,000}{(1.0225)^{40}}$

Now, let's compute the value of $(1.0225)^{40}$ and then solve for PV.

I’ll calculate that next.The amount you need to invest now in one lump sum to have $25,000 after 10 years with an APR of 9% compounded quarterly is approximately $10,266.14.

Would you like more details or have any further questions?

Here are 5 related questions to expand this topic:

1. How would the present value change if the APR were 7% instead of 9%?
2. What if the compounding were done monthly instead of quarterly?
3. How would the result differ if the investment period were 20 years instead of 10?
4. How is the formula affected by continuous compounding instead of quarterly?
5. What is the total interest earned over the 10-year period?

Tip: Increasing the number of compounding periods increases the final value due to the effect of more frequent interest calculations.