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Identify the explicit function for the sequence in the table.

[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
1 & 7 \\
\hline
2 & 19 \\
\hline
3 & 31 \\
\hline
4 & 43 \\
\hline
5 & 55 \\
\hline
\end{tabular}
\][/tex]

A. [tex]$a(n)=7+(n-1) \cdot 12$[/tex]
B. [tex]$a(n)=12+(n-1) \cdot 7$[/tex]
C. [tex]$a(n)=7(n-1)$[/tex]
D. [tex]$a(n)=12(n-1)$[/tex]

Sagot :

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]