To determine the number that belongs in the blank space for the recursive formula of the arithmetic sequence, we first need to understand both the explicit and recursive formulas for arithmetic sequences.
1. Explicit Formula:
The explicit formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is:
[tex]\[
a_n = 3 + (n-1) \cdot 8
\][/tex]
This formula gives the [tex]\(n\)[/tex]-th term directly in terms of [tex]\(n\)[/tex], where:
- 3 is the first term ([tex]\(a_1\)[/tex]) of the sequence.
- 8 is the common difference ([tex]\(d\)[/tex]) between consecutive terms.
2. Common Difference:
In an arithmetic sequence, the common difference ([tex]\(d\)[/tex]) is the constant amount by which each term differs from the previous term. This can be observed from the expression within the explicit formula:
[tex]\[
a_n = 3 + (n-1) \cdot 8
\][/tex]
The term [tex]\((n-1) \cdot 8\)[/tex] indicates that [tex]\(d = 8\)[/tex].
3. Recursive Formula:
The recursive formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence can be written as:
[tex]\[
a_n = a_{n-1} + d
\][/tex]
Since we identified the common difference ([tex]\(d\)[/tex]) from the explicit formula as 8, the recursive formula can be written as:
[tex]\[
a_n = a_{n-1} + 8
\][/tex]
4. Conclusion:
The number that belongs in the blank space in the recursive formula is:
[tex]\[
a_n = a_{n-1} + 8
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{8}
\][/tex]
