Complete the table to determine the effect of the number of compounding periods when computing interest. Suppose that \[tex]$10,000 is invested at 3.5% interest for 18 years 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 & $[/tex]n[tex]$ Value & Result \\
\hline
(a) & Annually & $[/tex]n=\square[tex]$ & $[/tex]\[tex]$ \square$[/tex] \\
\hline
(b) & Quarterly & [tex]$n=\square$[/tex] & [tex]$\$[/tex] \square[tex]$ \\
\hline
(c) & Monthly & $[/tex]n=\square[tex]$ & $[/tex]\[tex]$ \square$[/tex] \\
\hline
(d) & Daily & [tex]$n=365$[/tex] & [tex]$\$[/tex] \square[tex]$ \\
\hline
(e) & Continuously & Not Applicable & $[/tex]\[tex]$ \square$[/tex] \\
\hline
\end{tabular}



Answer :

Let's fill in the table step-by-step based on the given details about the investment of [tex]$\$[/tex]10,000[tex]$ at $[/tex]3.5\%[tex]$ interest for 18 years under various compounding options. First, we summarize the necessary formulas for each compounding method. ### Compounding Annually For annual compounding, the formula is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where \( P \) is the principal amount (\$[/tex]10,000), [tex]\( r \)[/tex] is the annual interest rate (0.035), [tex]\( n \)[/tex] is the number of times the interest is compounded per year (1 for annually), and [tex]\( t \)[/tex] is the time the money is invested for (18 years).

### Compounding Quarterly
For quarterly compounding, the formula is the same:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where [tex]\( n = 4 \)[/tex] (since interest is compounded quarterly).

### Compounding Monthly
For monthly compounding, the formula is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where [tex]\( n = 12 \)[/tex] (since interest is compounded monthly).

### Compounding Daily
For daily compounding, the formula is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where [tex]\( n = 365 \)[/tex] (since interest is compounded daily).

### Compounding Continuously
For continuous compounding, the formula is:
[tex]\[ A = P e^{rt} \][/tex]
Where [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).

Let's fill in the table with the results:

[tex]\[ \begin{tabular}{|l|l|c|c|} \hline & Compounding Option & n Value & Result \\ \hline (a) & Annually & n = 1 & \$ 18,574.89 \\ \hline (b) & Quarterly & n = 4 & \$ 18,724.72 \\ \hline (c) & Monthly & n = 12 & \$ 18,758.90 \\ \hline (d) & Daily & n = 365 & \$ 18,775.54 \\ \hline (e) & Continuously & Not Applicable & \$ 18,776.11 \\ \hline \end{tabular} \][/tex]

Here is a concise detailed explanation of the process used to arrive at each value:

1. Annually: The interest is compounded once per year (n = 1). The future value after 18 years is \[tex]$ 18,574.89. 2. Quarterly: The interest is compounded four times per year (n = 4). The future value after 18 years is \$[/tex] 18,724.72.
3. Monthly: The interest is compounded twelve times per year (n = 12). The future value after 18 years is \[tex]$ 18,758.90. 4. Daily: The interest is compounded daily (n = 365). The future value after 18 years is \$[/tex] 18,775.54.
5. Continuously: The interest is compounded continuously, meaning the number of compounding periods approaches infinity. The future value after 18 years is \$ 18,776.11.

This table clearly demonstrates the effect of increasing the frequency of compounding on the accumulated amount. As the compounding frequency increases, the future value slightly increases as well.