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Sagot :
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.
### 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.
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