Questions: Use the compound interest formulas A=P(1+r/n)^(nt) and A=Pe^(tt) to solve the problem given. Round answers to the nearest cent. Find the accumulated value of an investment of 20,000 for 4 years at an interest rate of 6.5% if the money is a. compounded semiannually; b. compounded quarterly, c. compounded monthly, d. compounded continuously. a. What is the accumulated value if the money is compounded semiannually? 25831.55 (Round your answer to the nearest cent. Do not include the symbol in your answer.) b. What is the accumulated value if the money is compounded quarterly? 25884.45 (Round your answer to the nearest cent. Do not include the symbol in your answer.) c. What is the accumulated value if the money is compounded monthly? (Round your answer to the nearest cent. Do not include the symbol in your answer.)

Use the compound interest formulas A=P(1+r/n)^(nt) and A=Pe^(tt) to solve the problem given. Round answers to the nearest cent.
Find the accumulated value of an investment of 20,000 for 4 years at an interest rate of 6.5% if the money is a. compounded semiannually; b. compounded quarterly, c. compounded monthly, d. compounded continuously.
a. What is the accumulated value if the money is compounded semiannually?
25831.55
(Round your answer to the nearest cent. Do not include the symbol in your answer.)
b. What is the accumulated value if the money is compounded quarterly?
25884.45
(Round your answer to the nearest cent. Do not include the symbol in your answer.)
c. What is the accumulated value if the money is compounded monthly?

(Round your answer to the nearest cent. Do not include the symbol in your answer.)

Transcript text: Use the compound interest formulas $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{r}}{\mathrm{n}}\right)^{\mathrm{nt}}$ and $\mathrm{A}=\mathrm{P} \mathrm{e}^{\mathrm{tt}}$ to solve the problem given. Round answers to the nearest cent. Find the accumulated value of an investment of $\$ 20,000$ for 4 years at an interest rate of $6.5 \%$ if the money is a. compounded semiannually; b. compounded quarterly, c. compounded monthly, d. compounded continuously. a. What is the accumulated value if the money is compounded semiannually? \$ 25831.55 (Round your answer to the nearest cent. Do not include the $\$$ symbol in your answer.) b. What is the accumulated value if the money is compounded quarterly? \$ 25884.45 (Round your answer to the nearest cent. Do not include the $\$$ symbol in your answer.) c. What is the accumulated value if the money is compounded monthly? \$ $\square$ (Round your answer to the nearest cent. Do not include the $\$$ symbol in your answer.)

Solution

Solution Steps

To solve the problem, we will use the compound interest formula for different compounding periods. For parts a, b, and c, we will use the formula $ A = P \left(1 + \frac{r}{n}\right)^{nt} $, where $ P $ is the principal amount, $ r $ is the annual interest rate, $ n $ is the number of times interest is compounded per year, and $ t $ is the number of years. For part d, we will use the formula $ A = P e^{rt} $ for continuous compounding.

Solution Approach

For part a, use $ n = 2 $ (semiannually).
For part b, use $ n = 4 $ (quarterly).
For part c, use $ n = 12 $ (monthly).

Step 1: Calculate Accumulated Value for Semiannual Compounding

Using the formula for compound interest, we have:

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

For semiannual compounding, where $ P = 20000 $, $ r = 0.065 $, $ n = 2 $, and $ t = 4 $:

\[ A = 20000 \left(1 + \frac{0.065}{2}\right)^{2 \cdot 4} = 20000 \left(1 + 0.0325\right)^{8} = 20000 \left(1.0325\right)^{8} \approx 25831.55 \]

Step 2: Calculate Accumulated Value for Quarterly Compounding

For quarterly compounding, where $ n = 4 $:

\[ A = 20000 \left(1 + \frac{0.065}{4}\right)^{4 \cdot 4} = 20000 \left(1 + 0.01625\right)^{16} = 20000 \left(1.01625\right)^{16} \approx 25884.45 \]

Step 3: Calculate Accumulated Value for Monthly Compounding

For monthly compounding, where $ n = 12 $:

\[ A = 20000 \left(1 + \frac{0.065}{12}\right)^{12 \cdot 4} = 20000 \left(1 + 0.00541667\right)^{48} = 20000 \left(1.00541667\right)^{48} \approx 25920.41 \]