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Use the compound interest formulas [tex]\( A = P\left(1 + \frac{r}{n}\right)^{nt} \)[/tex] and [tex]\( A = Pe^{rt} \)[/tex] to solve the problem given. Round answers to the nearest cent.

Find the accumulated value of an investment of [tex]\(\$ 20,000\)[/tex] for 4 years at an interest rate of [tex]\(6\%\)[/tex].

a. What is the accumulated value if the money is compounded semiannually?
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(Round your answer to the nearest cent. Do not include the [tex]\(\$ \)[/tex] symbol in your answer.)

b. What is the accumulated value if the money is compounded quarterly?
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(Round your answer to the nearest cent. Do not include the [tex]\(\$ \)[/tex] symbol in your answer.)

c. What is the accumulated value if the money is compounded monthly?
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(Round your answer to the nearest cent. Do not include the [tex]\(\$ \)[/tex] symbol in your answer.)

d. What is the accumulated value if the money is compounded continuously?
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(Round your answer to the nearest cent. Do not include the [tex]\(\$ \)[/tex] symbol in your answer.)



Answer :

To solve this problem, let's write down the formulas and the values we have:

- Principal (P): \[tex]$20,000 - Annual interest rate (r): \(6\% = 0.06\) - Time (t): 4 years ### a. Compounded Semiannually For interest compounded semiannually, the number of compounding periods per year (n) is 2. The formula to use is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Substituting the values: \[ A = 20000 \left(1 + \frac{0.06}{2}\right)^{2 \times 4} \] \[ A = 20000 \left(1 + 0.03\right)^8 \] \[ A = 20000 \left(1.03\right)^8 \] \[ A \approx 20000 \times 1.26677 \] \[ A \approx 25335.4 \] So, the accumulated value if the money is compounded semiannually is \$[/tex]25,335.40.

### b. Compounded Quarterly

For interest compounded quarterly, the number of compounding periods per year (n) is 4. The formula to use is:

[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]

Substituting the values:

[tex]\[ A = 20000 \left(1 + \frac{0.06}{4}\right)^{4 \times 4} \][/tex]
[tex]\[ A = 20000 \left(1 + 0.015\right)^{16} \][/tex]
[tex]\[ A = 20000 \left(1.015\right)^{16} \][/tex]
[tex]\[ A \approx 20000 \times 1.2689855 \][/tex]
[tex]\[ A \approx 25379.71 \][/tex]

So, the accumulated value if the money is compounded quarterly is \[tex]$25,379.71. ### c. Compounded Monthly For interest compounded monthly, the number of compounding periods per year (n) is 12. The formula to use is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Substituting the values: \[ A = 20000 \left(1 + \frac{0.06}{12}\right)^{12 \times 4} \] \[ A = 20000 \left(1 + 0.005\right)^{48} \] \[ A = 20000 \left(1.005\right)^{48} \] \[ A \approx 20000 \times 1.270389 \] \[ A \approx 25409.78 \] So, the accumulated value if the money is compounded monthly is \$[/tex]25,409.78.