To convert the arithmetic sequence given in explicit form to a recursive form, let's break this down step-by-step:
1. Understand the given explicit formula:
The explicit form of the arithmetic sequence is given by the formula:
[tex]\[
a_n = -8 + 2(n - 1)
\][/tex]
This formula directly generates the [tex]\(n\)[/tex]-th term of the sequence.
2. Identify the base term [tex]\(a_1\)[/tex]:
To find the first term [tex]\(a_1\)[/tex] of this sequence, set [tex]\(n = 1\)[/tex]:
[tex]\[
a_1 = -8 + 2(1 - 1) = -8 + 2 \cdot 0 = -8
\][/tex]
So, [tex]\(a_1 = -8\)[/tex].
3. Determine the common difference:
The common difference [tex]\(d\)[/tex] in an arithmetic sequence is the difference between any two successive terms. From the given formula, we identify the common difference as the coefficient of [tex]\(n\)[/tex], which is [tex]\(2\)[/tex].
4. Construct the recursive formula:
A recursive formula provides the [tex]\(n\)[/tex]-th term of the sequence in terms of the previous term. For an arithmetic sequence with common difference [tex]\(d\)[/tex], the recursive formula is typically:
[tex]\[
a_n = a_{n-1} + d
\][/tex]
Given our common difference [tex]\(d = 2\)[/tex], the recursive formula becomes:
[tex]\[
a_n = a_{n-1} + 2
\][/tex]
5. Include the initial term:
The initial term, as previously found, is [tex]\(a_1 = -8\)[/tex].
So, putting it all together, the recursive form of the given arithmetic sequence is:
[tex]\[
a_1 = -8
\][/tex]
[tex]\[
a_n = a_{n-1} + 2 \quad \text{for} \quad n \geq 2
\][/tex]
Therefore, the correct choice from the given options is:
[tex]\[
a_1 = -8, \quad a_n = a_{n-1} + 2
\][/tex]
