Use the compound interest formulas $A=P\left(1+\frac{r}{n}\right)^{n t}$ and $A=P e^{r t}$ to solve the problem given. Round answers to the nearest cent. Find the accumulated value of an investment of $\$ 20,000$ for 3 years at an interest rate of $6 \%$ if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; $d$. compounded continuously.

Step 1 ：Given the principal amount (P) as $20000, the interest rate (r) as $0.06, and the number of years the money is invested for (t) as 3, we can use these values in the compound interest formula to calculate the accumulated value (A) for each compounding frequency. For the continuously compounded case, we use the continuous compounding formula. The interest rate should be converted from percentage to a decimal before using in the formulas.

Step 2 ：For semiannual compounding (n=2), we use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to get \(A_{semiannually} = 23881.05\).

Step 3 ：For quarterly compounding (n=4), we use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to get \(A_{quarterly} = 23912.36\).

Step 4 ：For monthly compounding (n=12), we use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to get \(A_{monthly} = 23933.61\).

Step 5 ：For continuous compounding, we use the formula \(A=Pe^{rt}\) to get \(A_{continuously} = 23944.35\).

Step 6 ：Final Answer: The accumulated value of the investment is \(\boxed{\$23881.05}\) when compounded semiannually, \(\boxed{\$23912.36}\) when compounded quarterly, \(\boxed{\$23933.61}\) when compounded monthly, and \(\boxed{\$23944.35}\) when compounded continuously.