To identify the explicit function for the sequence in the table, we need to analyze the given values and determine the pattern. Let's examine the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 8 \\
\hline
2 & 14 \\
\hline
3 & 20 \\
\hline
4 & 26 \\
\hline
5 & 32 \\
\hline
\end{array}
\][/tex]
First, let's identify the common difference in the sequence of [tex]\( y \)[/tex] values:
[tex]\[
\begin{aligned}
14 - 8 &= 6, \\
20 - 14 &= 6, \\
26 - 20 &= 6, \\
32 - 26 &= 6.
\end{aligned}
\][/tex]
We see that the difference between successive terms is constant and equals 6. This indicates an arithmetic sequence where the common difference [tex]\( d \)[/tex] is 6.
Next, we use the explicit formula for an arithmetic sequence:
[tex]\[
a(n) = a_1 + (n-1) \cdot d,
\][/tex]
where [tex]\( a_1 \)[/tex] is the first term and [tex]\( d \)[/tex] is the common difference.
In this case:
- The first term [tex]\( a_1 = 8 \)[/tex].
- The common difference [tex]\( d = 6 \)[/tex].
Substituting these values into the formula, we get:
[tex]\[
a(n) = 8 + (n-1) \cdot 6.
\][/tex]
So, the explicit function for the sequence is:
[tex]\[
a(n) = 8 + (n-1) \cdot 6.
\][/tex]
Therefore, the correct answer is:
B. [tex]\( a(n) = 8 + (n-1) \cdot 6 \)[/tex].
