To identify the explicit function for the given sequence, we can follow a step-by-step approach to find a linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Let's analyze the data points provided in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 7 \\
\hline
2 & 19 \\
\hline
3 & 31 \\
\hline
4 & 43 \\
\hline
5 & 55 \\
\hline
\end{array}
\][/tex]
We observe the formation of a linear relationship since [tex]\( y \)[/tex] changes by a consistent amount as [tex]\( x \)[/tex] increases. Specifically, as [tex]\( x \)[/tex] increases by 1, [tex]\( y \)[/tex] increases by 12.
### Step-by-Step Solution:
1. Determine the slope (rate of change):
- Observe that as [tex]\( x \)[/tex] increases from 1 to 2, [tex]\( y \)[/tex] changes from 7 to 19. The change in [tex]\( y \)[/tex] is [tex]\( 19 - 7 = 12 \)[/tex].
- Similarly, as [tex]\( x \)[/tex] changes from 2 to 3, [tex]\( y \)[/tex] changes from 19 to 31. Again, the difference is [tex]\( 31 - 19 = 12 \)[/tex].
- Continuing this analysis, we find the consistent rate of change is 12.
Thus, the slope [tex]\( m \)[/tex] of the linear function is 12.
2. Identify the intersection (y-intercept):
- The y-intercept is found when [tex]\( n = 1 \)[/tex]. In this case, [tex]\( y = 7 \)[/tex].
3. Formulate the explicit linear function:
- Using the point-slope form [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope, and [tex]\( c \)[/tex] is the y-intercept.
- Given that the intercept in context of [tex]\( n = 1 \)[/tex] yields [tex]\( y = 12*(n-1) + 7 \)[/tex].
Hence, the explicit function is of the pattern [tex]\( y = 7 + 12(n-1) \)[/tex].
Checking the options:
- Option A: [tex]\( y = 7(n-1) \)[/tex]
- Option B: [tex]\( y = 12 + (n-1) \cdot 7 \)[/tex]
- Option C: [tex]\( y = 12(n-1) \)[/tex]
- Option D: [tex]\( y = 7 + (n-1) \cdot 12 \)[/tex]
The option that matches the identified explicit function of [tex]\( y = 7 + 12(n-1) \)[/tex] is:
[tex]\[
\boxed{D}
\][/tex]
Therefore, the correct explicit function is given by option [tex]\( D \)[/tex]: [tex]\( y = 7 + (n-1) \cdot 12 \)[/tex].
