To identify the explicit function for the sequence provided, let’s analyze the given values and formulate the rule step-by-step.
The sequence provided in the table is:
[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]
1. Check the differences between successive terms:
- [tex]\(19 - 7 = 12\)[/tex]
- [tex]\(31 - 19 = 12\)[/tex]
- [tex]\(43 - 31 = 12\)[/tex]
- [tex]\(55 - 43 = 12\)[/tex]
The difference between successive [tex]\( y \)[/tex]-values is consistent, indicating that the sequence is arithmetic with a common difference of 12.
2. Determine the general formula for the arithmetic sequence:
The general formula for an arithmetic sequence is:
[tex]\[
a(n) = a_1 + (n - 1) \cdot d
\][/tex]
where:
- [tex]\(a_1\)[/tex] is the first term,
- [tex]\(d\)[/tex] is the common difference, and
- [tex]\(n\)[/tex] is the term number.
3. Identify the first term [tex]\(a_1\)[/tex]:
From the table, when [tex]\(x = 1\)[/tex],
[tex]\[
a_1 = 7
\][/tex]
4. Identify the common difference [tex]\(d\)[/tex]:
From the differences calculated earlier,
[tex]\[
d = 12
\][/tex]
5. Substitute [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex] into the formula:
[tex]\[
a(n) = 7 + (n - 1) \cdot 12
\][/tex]
Hence, the explicit formula for the sequence is:
[tex]\[
a(n) = 7 + (n - 1) \cdot 12
\][/tex]
Reviewing the given options:
- [tex]\(A. a(n) = 7 + (n - 1) \cdot 12\)[/tex]
- [tex]\(B. a(n) = 12 + (n - 1) \cdot 7\)[/tex]
- [tex]\(C. a(n) = 7(n - 1)\)[/tex]
- [tex]\(D. a(n) = 12(n - 1)\)[/tex]
The correct formula corresponds to option [tex]\(A\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
