find the accumulated value of an investment of 20,000 for 3 years at an interest rate of 7%, if the money is a compounded semiannually; b. coumpounded quaterly; c. compounded monthly d. coumpounded continuously.

To find the accumulated value of an investment, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = Accumulated value
P = Principal (initial investment)
r = Annual interest rate (as a decimal)
n = Number of times the interest is compounded per year
t = Number of years

For each scenario, we will substitute the given values into the formula.

a) Compounded semiannually:
r = 7% = 0.07 (as a decimal)
n = 2 (since interest is compounded semiannually)
t = 3 years

A = 20,000(1 + 0.07/2)^(2*3)

Now, we can calculate A:
A = 20,000(1 + 0.035)^6

Calculating further,
A = 20,000(1.035)^6
A ≈ 24,065.39

Therefore, the accumulated value of the investment after 3 years, compounded semiannually, is approximately $24,065.39.

b) Compounded quarterly:
r = 7% = 0.07 (as a decimal)
n = 4 (since interest is compounded quarterly)
t = 3 years

A = 20,000(1 + 0.07/4)^(4*3)

Now, we can calculate A:
A = 20,000(1 + 0.0175)^12

Calculating further,
A = 20,000(1.0175)^12
A ≈ 24,253.08

Therefore, the accumulated value of the investment after 3 years, compounded quarterly, is approximately $24,253.08.

c) Compounded monthly:
r = 7% = 0.07 (as a decimal)
n = 12 (since interest is compounded monthly)
t = 3 years

A = 20,000(1 + 0.07/12)^(12*3)

Now, we can calculate A:
A = 20,000(1 + 0.00583)^36

Calculating further,
A = 20,000(1.00583)^36
A ≈ 24,379.41

Therefore, the accumulated value of the investment after 3 years, compounded monthly, is approximately $24,379.41.

d) Compounded continuously:
r = 7% = 0.07 (as a decimal)
n is no longer needed in the formula since interest is compounded continuously
t = 3 years

A = 20,000e^(0.07*3)

Now, we can calculate A:
A = 20,000e^(0.21)

Calculating further,
A ≈ 20,000 * 1.23203
A ≈ 24,640.60

Therefore, the accumulated value of the investment after 3 years, compounded continuously, is approximately $24,640.60.