To identify the explicit function for the sequence given in the table, we need to observe the pattern in the values of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
1 & 9 \\
\hline
2 & 14 \\
\hline
3 & 19 \\
\hline
4 & 24 \\
\hline
5 & 29 \\
\hline
\end{tabular}
\][/tex]
First, let's see the changes in the [tex]\( y \)[/tex]-values as [tex]\( x \)[/tex] increases:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 9 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 14 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 19 \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( y = 24 \)[/tex]
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 29 \)[/tex]
We notice that each time [tex]\( x \)[/tex] increases by 1, [tex]\( y \)[/tex] increases by 5. This suggests a linear relationship of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
The slope [tex]\( m \)[/tex] (or the common difference in an arithmetic sequence) is 5 because:
[tex]\[
14 - 9 = 5, \quad 19 - 14 = 5, \quad 24 - 19 = 5, \quad 29 - 24 = 5
\][/tex]
Now we need to find the y-intercept ([tex]\( b \)[/tex]) of the linear function. For [tex]\( x = 1 \)[/tex], [tex]\( y = 9 \)[/tex]. Using the linear equation form:
[tex]\[
y = mx + b
\][/tex]
[tex]\[
9 = 5(1) + b
\][/tex]
[tex]\[
9 = 5 + b
\][/tex]
[tex]\[
b = 4
\][/tex]
However, based on the structure of the options provided, we should express the function with an offset form starting from [tex]\( y \)[/tex] when [tex]\( x = 1 \)[/tex].
By observing the table carefully and analyzing the pattern:
For the sequence, the correct explicit formula is:
[tex]\[
a(n) = 9 + (n-1) \cdot 5
\][/tex]
To confirm this, we check:
- For [tex]\( x = 1 \)[/tex]: [tex]\( 9 + (1-1) \cdot 5 = 9 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( 9 + (2-1) \cdot 5 = 14 \)[/tex]
- For [tex]\( x = 3 \)[/tex]: [tex]\( 9 + (3-1) \cdot 5 = 19 \)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\( 9 + (4-1) \cdot 5 = 24 \)[/tex]
- For [tex]\( x = 5 \)[/tex]: [tex]\( 9 + (5-1) \cdot 5 = 29 \)[/tex]
Thus, the final explicit function that matches the pattern in the table is:
B. [tex]\( a(n) = 9 + (n-1) \cdot 5 \)[/tex]
