Questions: Complete the table to determine the effect of the number of compounding periods when computing interest. Suppose that 8000 is invested at 3.5 % interest for 17 yr under the following compounding options. Round answers in the second column to the nearest whole number. Round answers in the last column to the nearest cent. - (a) Annually n= - (b) Quarterly n= - (c) Monthly n= - (d) Daily n=365 - (e) Continuously Not Applicable

Complete the table to determine the effect of the number of compounding periods when computing interest. Suppose that 8000 is invested at 3.5 % interest for 17 yr under the following compounding options. Round answers in the second column to the nearest whole number. Round answers in the last column to the nearest cent.

- (a) Annually n=
- (b) Quarterly n=
- (c) Monthly n=
- (d) Daily n=365
- (e) Continuously Not Applicable

Transcript text: Complete the table to determine the effect of the number of compounding periods when computing interest. Suppose that $\$ 8000$ is invested at $3.5 \%$ interest for 17 yr under the following compounding options. Round answers in the second column to the nearest whole number. Round answers in the last column to the nearest cent. \begin{tabular}{|l|l|c|c|} \hline & Compounding Option & $n$ Value & Result \\ \hline (a) Annually & $n=\square$ & $\$ \square$ \\ \hline (b) Quarterly & $n=\square$ & $\$ \square$ \\ \hline (c) Monthly & $n=\square$ & $\$ \square$ \\ \hline (d) Daily & $n=365$ & $\$ \square$ \\ \hline (e) Continuously & Not Applicable & $\$ \square$ \\ \hline \end{tabular}

Solution

Solution Steps

To solve this problem, we need to calculate the future value of an investment using different compounding options. For each option, we will use the formula for compound interest: $ A = P \left(1 + \frac{r}{n}\right)^{nt} $, where $ A $ is the amount of money accumulated after n years, including interest, $ P $ is the principal amount (\$8000), $ r $ is the annual interest rate (3.5% or 0.035), $ n $ is the number of times that interest is compounded per year, and $ t $ is the time the money is invested for (17 years). For continuous compounding, we use the formula $ A = Pe^{rt} $.

Step 1: Calculate Future Value for Annual Compounding

For annual compounding ($ n = 1 $), the future value $ A $ is calculated as follows: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} = 8000 \left(1 + \frac{0.035}{1}\right)^{1 \cdot 17} = 8000 \left(1.035\right)^{17} \approx 14357.4 \]

Step 2: Calculate Future Value for Quarterly Compounding

For quarterly compounding ($ n = 4 $), the future value $ A $ is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} = 8000 \left(1 + \frac{0.035}{4}\right)^{4 \cdot 17} = 8000 \left(1.00875\right)^{68} \approx 14466.76 \]

Step 3: Calculate Future Value for Monthly Compounding

For monthly compounding ($ n = 12 $), the future value $ A $ is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} = 8000 \left(1 + \frac{0.035}{12}\right)^{12 \cdot 17} = 8000 \left(1.00291667\right)^{204} \approx 14491.69 \]

Step 4: Calculate Future Value for Daily Compounding

For daily compounding ($ n = 365 $), the future value $ A $ is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} = 8000 \left(1 + \frac{0.035}{365}\right)^{365 \cdot 17} = 8000 \left(1.00009589\right)^{6205} \approx 14503.83 \]

Step 5: Calculate Future Value for Continuous Compounding

For continuous compounding, the future value $ A $ is calculated using: \[ A = Pe^{rt} = 8000 e^{0.035 \cdot 17} \approx 14504.25 \]

Final Answer

The results for each compounding option are as follows:

(a) Annually: $ A \approx 14357.4 $
(b) Quarterly: $ A \approx 14466.76 $
(c) Monthly: $ A \approx 14491.69 $
(d) Daily: $ A \approx 14503.83 $
(e) Continuously: $ A \approx 14504.25 $

Thus, the final answers are: \[ \boxed{A_{\text{annually}} \approx 14357.4} \] \[ \boxed{A_{\text{quarterly}} \approx 14466.76} \] \[ \boxed{A_{\text{monthly}} \approx 14491.69} \]

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