To find the explicit formula for an arithmetic sequence given its recursive definition, let’s start by identifying the key components:
1. First term ([tex]\(a_1\)[/tex]): This is provided directly by the problem.
[tex]\[ a_1 = 8 \][/tex]
2. Common difference ([tex]\(d\)[/tex]): This is the amount by which each term increases (or decreases) from the previous term. For this sequence:
[tex]\[ a_n = a_{n-1} - 6 \][/tex]
This means each term decreases by 6. Therefore, the common difference is:
[tex]\[ d = -6 \][/tex]
The formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence (explicit formula) is:
[tex]\[ a_n = a_1 + (n-1) \cdot d \][/tex]
Now, substitute the known values into this formula:
- [tex]\(a_1 = 8\)[/tex]
- [tex]\(d = -6\)[/tex]
[tex]\[ a_n = 8 + (n-1) \cdot (-6) \][/tex]
Now let's simplify and verify that this matches one of the provided options:
[tex]\[ a_n = 8 + (n-1) \cdot (-6) \][/tex]
[tex]\[ a_n = 8 + (n-1)(-6) \][/tex]
This corresponds directly to option B:
[tex]\[ \text{B. } a_n = 8 + (n-1) \cdot (-6) \][/tex]
Therefore, the explicit formula for this arithmetic sequence is:
[tex]\[ \boxed{a_n = 8 + (n-1) \cdot (-6)} \][/tex]
And the correct choice is:
[tex]\[ \boxed{\text{B}} \][/tex]
