Suppose $$ \$ 10,000 $$ is invested by Stephanie for 25 years in an account that earns $$ 12 \% $$ interest, compounded semi-annually. Round the solutions to the nearest cent, if necessary. Determine the future value of the account. Future Value $$ = $$ Determine the amount of interest earned in this account over the 25 years. Interest $$ = $$

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We begin by using the compound interest formula

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

where

- $$P = 10\,000$$ is the principal,
- $$r = 0.12$$ is the annual interest rate (in decimal),
- $$n = 2$$ is the number of compounding periods per year (semi-annually),
- $$t = 25$$ years is the time period.

**Step 1. Calculate the Future Value**

Substitute the given values into the formula:

$$
A = 10\,000 \left(1 + \frac{0.12}{2}\right)^{2 \times 25} = 10\,000 \left(1 + 0.06\right)^{50} = 10\,000 \left(1.06\right)^{50}.
$$

We compute $$\left(1.06\right)^{50}$$. Using a calculator, we obtain

$$
\left(1.06\right)^{50} \approx 18.42068.
$$

Thus, the future value is

$$
A \approx 10\,000 \times 18.42068 = 184\,206.80.
$$

Rounded to the nearest cent, the future value is

$$
\text{Future Value} = \$184\,206.80.
$$

**Step 2. Calculate the Interest Earned**

The interest earned is the difference between the future value and the original principal:

$$
\text{Interest} = A - P \approx 184\,206.80 - 10\,000 = 174\,206.80.
$$

Rounded to the nearest cent, the interest earned is

$$
\text{Interest} = \$174\,206.80.
$$
**Future Value:** \$184,206.80 **Interest Earned:** \$174,206.80

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