an individual purchased a 3 - year, $10,000 promissory note with an interest rate of 4.5%/year compounded…

Explanation:

Step1: Identify the compound - interest formula

The formula for compound interest is (A = P(1+\frac{r}{n})^{nt}), where (A) is the future value, (P) is the principal (initial amount), (r) is the annual interest rate (in decimal), (n) is the number of times compounded per year, and (t) is the number of years. We want to find (P), so we can rewrite the formula as (P=\frac{A}{(1 +\frac{r}{n})^{nt}}). Given (A=$10000), (r = 0.045) (since (4.5%=0.045)), (n = 2) (compounded semiannually), and (t = 3) years.

Step2: Calculate the value of ((1+\frac{r}{n})^{nt})

First, calculate (\frac{r}{n}=\frac{0.045}{2}=0.0225) and (nt=2\times3 = 6). Then ((1+\frac{r}{n})^{nt}=(1 + 0.0225)^{6}). Using the formula ((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}) or a calculator, ((1.0225)^{6}\approx1.142825).

Step3: Calculate the principal (P)

Substitute into the formula (P=\frac{A}{(1+\frac{r}{n})^{nt}}), so (P=\frac{10000}{1.142825}\approx8750.29).

Answer:

(8750.29)