Evaluate $\$20,000\left[\frac{{\left(1+\frac{0.05}{2}\right)}^4-1}{\frac{0.05}{2}}\right]$

A) $172,405.00
B) $82,207.63
C) $83,050.31
D) $41,525.16
E) $86,202.50

The expression is a formula for the future value of a series of cash flows (like an annuity). The formula is used to calculate the future value of an annuity due to compound interest. Here, the interest rate per period is 0.05/2 = 0.025 (since it's compounded semi-annually), and the number of periods is 4. Plugging these values into the formula gives: $\$20,000\left[\frac{{\left(1+0.025\right)}^4-1}{0.025}\right]$ Calculating the expression inside the brackets first: ${\left(1+0.025\right)}^4-1=(1.025{)}^4-1\approx 0.103812890625$ Then, dividing by 0.025: $\frac{0.103812890625}{0.025}\approx 4.152515625$ Finally, multiplying by \$20,000: $\$20,000\times 4.152515625\approx \$83,050.31$ Therefore, the correct answer is \$83,050.31, which is option C.