We are asked to find the present value $P$ of an investment that will amount to $250,000 after 10 years, if it is invested at an annual interest rate of 6% compounded semiannually.

1. Problem Description

We are asked to find the present value $P$ of an investment that will amount to $250,000 after 10 years, if it is invested at an annual interest rate of 6% compounded semiannually.

2. Solution Steps

The formula for the future value $FV$ of an investment with present value $P$, interest rate $r$ per compounding period, and $n$ compounding periods is:

$FV = P(1 + r)^n$

In this problem, we are given $FV = 250000$, the annual interest rate is 6%, so the interest rate per semiannual period is $r = 0.06/2 = 0.03$. The investment is for 10 years, and it is compounded semiannually, so the number of compounding periods is $n = 10 \times 2 = 20$.

We want to find the present value $P$. We can rearrange the formula to solve for $P$:

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

Plugging in the values we have:

$P = \frac{250000}{(1 + 0.03)^{20}}$

$P = \frac{250000}{(1.03)^{20}}$

$P = \frac{250000}{1.80611123466702}$

$P \approx 138418.7749$

Rounding to the nearest cent, we get $P = 138418.77$.

3. Final Answer

P = \138418.77$