Ise the compound interest formulas $$ A=P\left(1+\frac{r}{n}\right)^{n t} $$ and $$ A=P e^{n t} $$ to solve the problem given. Round answers to the nearest cent. ind the accumulated value of an investment of $$ \$ 25,000 $$ for 6 years at an interest rate of $$ 4 \% $$ if the money is a. compounded semiannually; b. compounded quarterly; ompounded monthly; $$ d $$. compounded continuously. What is the accumulated value if the money is compounded semiannually? Round your answer to the nearest cent. Do not include the $$ \$ $$ symbol in vour answer)

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We start with the compound interest formula for periodic compounding:

$$
A = P\left(1 + \frac{r}{n}\right)^{n t}
$$

For semiannual compounding, the number of compounding periods per year is $$ n = 2 $$.

Given:
- Principal: $$ P = 25000 $$
- Annual interest rate: $$ r = 0.04 $$
- Number of years: $$ t = 6 $$

Substitute the values into the formula:

$$
A = 25000\left(1 + \frac{0.04}{2}\right)^{2 \times 6}
$$

Simplify inside the parentheses:

$$
1 + \frac{0.04}{2} = 1 + 0.02 = 1.02
$$

The exponent is:

$$
2 \times 6 = 12
$$

Thus, the formula becomes:

$$
A = 25000 \times (1.02)^{12}
$$

Next, calculate $$(1.02)^{12}$$. Using a calculator:

$$
(1.02)^{12} \approx 1.26824
$$

Multiply by the principal:

$$
A \approx 25000 \times 1.26824 \approx 31706.03
$$

Rounded to the nearest cent, the accumulated value is:

$$
31706.03
$$
The accumulated value is \$31,706.03.

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