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].
