To determine the explicit rule for the given arithmetic sequence [tex]\(2, 9, 16, 23, 30, \ldots\)[/tex], we can follow these steps:
1. Identify the first term [tex]\(a_1\)[/tex]:
- The first term [tex]\(a_1\)[/tex] is [tex]\(2\)[/tex].
2. Find the common difference [tex]\(d\)[/tex]:
- The common difference [tex]\(d\)[/tex] can be found by subtracting the first term from the second term:
[tex]\[
d = 9 - 2 = 7
\][/tex]
- To verify, check the difference between consecutive terms:
[tex]\[
16 - 9 = 7, \quad 23 - 16 = 7, \quad 30 - 23 = 7
\][/tex]
- The common difference is consistently [tex]\(7\)[/tex].
3. Formulate the explicit rule of the arithmetic sequence:
- The general formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1)d
\][/tex]
- Substituting the values of [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[
a_n = 2 + (n - 1) \cdot 7
\][/tex]
- Simplify the expression:
[tex]\[
a_n = 2 + 7n - 7
\][/tex]
[tex]\[
a_n = 7n - 5
\][/tex]
Therefore, the explicit rule for the sequence [tex]\(2, 9, 16, 23, 30, \ldots\)[/tex] is [tex]\(a_n = 7n - 5\)[/tex].
Among the given options, the correct answer is:
[tex]\[
a_n = -5 + 7n
\][/tex]
This can also be written in the form provided as [tex]\(\boxed{a_n = -5 + 7n}\)[/tex].
