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To determine which of the provided equations accurately describes the function [tex]\( F(n) \)[/tex] given by the table of values, we will evaluate each equation for the given [tex]\( n \)[/tex] values and compare the results with the [tex]\( F(n) \)[/tex] values from the table.
We have the following [tex]\( n \)[/tex] values and [tex]\( F(n) \)[/tex] values from the table:
[tex]\[ \begin{array}{|c|c|} \hline n & F(n) \\ \hline 0 & 5 \\ \hline 1 & 8 \\ \hline 2 & 11 \\ \hline 5 & 20 \\ \hline 10 & 35 \\ \hline 20 & 65 \\ \hline \end{array} \][/tex]
Let's evaluate each of the function definitions:
### Option A: [tex]\( F(n) = 4n \)[/tex]
By substituting the [tex]\( n \)[/tex] values into the equation [tex]\( F(n) = 4n \)[/tex]:
[tex]\[ \begin{aligned} F(0) &= 4 \times 0 = 0 \\ F(1) &= 4 \times 1 = 4 \\ F(2) &= 4 \times 2 = 8 \\ F(5) &= 4 \times 5 = 20 \\ F(10) &= 4 \times 10 = 40 \\ F(20) &= 4 \times 20 = 80 \\ \end{aligned} \][/tex]
Comparing these results with the table:
[tex]\[ \begin{array}{c|c|c} n & F(n) & 4n \\ 0 & 5 & 0 \\ 1 & 8 & 4 \\ 2 & 11 & 8 \\ 5 & 20 & 20 \\ 10 & 35 & 40 \\ 20 & 65 & 80 \\ \end{array} \][/tex]
The values do not match, so [tex]\( F(n) = 4n \)[/tex] is not the correct equation.
### Option B: [tex]\( F(n) = n + 5 \)[/tex]
By substituting the [tex]\( n \)[/tex] values into the equation [tex]\( F(n) = n + 5 \)[/tex]:
[tex]\[ \begin{aligned} F(0) &= 0 + 5 = 5 \\ F(1) &= 1 + 5 = 6 \\ F(2) &= 2 + 5 = 7 \\ F(5) &= 5 + 5 = 10 \\ F(10) &= 10 + 5 = 15 \\ F(20) &= 20 + 5 = 25 \\ \end{aligned} \][/tex]
Comparing these results with the table:
[tex]\[ \begin{array}{c|c|c} n & F(n) & n + 5 \\ 0 & 5 & 5 \\ 1 & 8 & 6 \\ 2 & 11 & 7 \\ 5 & 20 & 10 \\ 10 & 35 & 15 \\ 20 & 65 & 25 \\ \end{array} \][/tex]
The values do not match, so [tex]\( F(n) = n + 5 \)[/tex] is not the correct equation.
### Option C: [tex]\( F(n) = 3n + 5 \)[/tex]
By substituting the [tex]\( n \)[/tex] values into the equation [tex]\( F(n) = 3n + 5 \)[/tex]:
[tex]\[ \begin{aligned} F(0) &= 3 \times 0 + 5 = 5 \\ F(1) &= 3 \times 1 + 5 = 8 \\ F(2) &= 3 \times 2 + 5 = 11 \\ F(5) &= 3 \times 5 + 5 = 20 \\ F(10) &= 3 \times 10 + 5 = 35 \\ F(20) &= 3 \times 20 + 5 = 65 \\ \end{aligned} \][/tex]
Comparing these results with the table:
[tex]\[ \begin{array}{c|c|c} n & F(n) & 3n + 5 \\ 0 & 5 & 5 \\ 1 & 8 & 8 \\ 2 & 11 & 11 \\ 5 & 20 & 20 \\ 10 & 35 & 35 \\ 20 & 65 & 65 \\ \end{array} \][/tex]
The values match perfectly, so [tex]\( F(n) = 3n + 5 \)[/tex] is the correct equation.
Therefore, the equation that describes [tex]\( F(n) \)[/tex] is Option C: [tex]\( F(n) = 3n + 5 \)[/tex].
We have the following [tex]\( n \)[/tex] values and [tex]\( F(n) \)[/tex] values from the table:
[tex]\[ \begin{array}{|c|c|} \hline n & F(n) \\ \hline 0 & 5 \\ \hline 1 & 8 \\ \hline 2 & 11 \\ \hline 5 & 20 \\ \hline 10 & 35 \\ \hline 20 & 65 \\ \hline \end{array} \][/tex]
Let's evaluate each of the function definitions:
### Option A: [tex]\( F(n) = 4n \)[/tex]
By substituting the [tex]\( n \)[/tex] values into the equation [tex]\( F(n) = 4n \)[/tex]:
[tex]\[ \begin{aligned} F(0) &= 4 \times 0 = 0 \\ F(1) &= 4 \times 1 = 4 \\ F(2) &= 4 \times 2 = 8 \\ F(5) &= 4 \times 5 = 20 \\ F(10) &= 4 \times 10 = 40 \\ F(20) &= 4 \times 20 = 80 \\ \end{aligned} \][/tex]
Comparing these results with the table:
[tex]\[ \begin{array}{c|c|c} n & F(n) & 4n \\ 0 & 5 & 0 \\ 1 & 8 & 4 \\ 2 & 11 & 8 \\ 5 & 20 & 20 \\ 10 & 35 & 40 \\ 20 & 65 & 80 \\ \end{array} \][/tex]
The values do not match, so [tex]\( F(n) = 4n \)[/tex] is not the correct equation.
### Option B: [tex]\( F(n) = n + 5 \)[/tex]
By substituting the [tex]\( n \)[/tex] values into the equation [tex]\( F(n) = n + 5 \)[/tex]:
[tex]\[ \begin{aligned} F(0) &= 0 + 5 = 5 \\ F(1) &= 1 + 5 = 6 \\ F(2) &= 2 + 5 = 7 \\ F(5) &= 5 + 5 = 10 \\ F(10) &= 10 + 5 = 15 \\ F(20) &= 20 + 5 = 25 \\ \end{aligned} \][/tex]
Comparing these results with the table:
[tex]\[ \begin{array}{c|c|c} n & F(n) & n + 5 \\ 0 & 5 & 5 \\ 1 & 8 & 6 \\ 2 & 11 & 7 \\ 5 & 20 & 10 \\ 10 & 35 & 15 \\ 20 & 65 & 25 \\ \end{array} \][/tex]
The values do not match, so [tex]\( F(n) = n + 5 \)[/tex] is not the correct equation.
### Option C: [tex]\( F(n) = 3n + 5 \)[/tex]
By substituting the [tex]\( n \)[/tex] values into the equation [tex]\( F(n) = 3n + 5 \)[/tex]:
[tex]\[ \begin{aligned} F(0) &= 3 \times 0 + 5 = 5 \\ F(1) &= 3 \times 1 + 5 = 8 \\ F(2) &= 3 \times 2 + 5 = 11 \\ F(5) &= 3 \times 5 + 5 = 20 \\ F(10) &= 3 \times 10 + 5 = 35 \\ F(20) &= 3 \times 20 + 5 = 65 \\ \end{aligned} \][/tex]
Comparing these results with the table:
[tex]\[ \begin{array}{c|c|c} n & F(n) & 3n + 5 \\ 0 & 5 & 5 \\ 1 & 8 & 8 \\ 2 & 11 & 11 \\ 5 & 20 & 20 \\ 10 & 35 & 35 \\ 20 & 65 & 65 \\ \end{array} \][/tex]
The values match perfectly, so [tex]\( F(n) = 3n + 5 \)[/tex] is the correct equation.
Therefore, the equation that describes [tex]\( F(n) \)[/tex] is Option C: [tex]\( F(n) = 3n + 5 \)[/tex].
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