To solve this problem, let's follow these steps carefully to derive the explicit formula for the arithmetic sequence given in the table and identify any domain restrictions.
### Step-by-Step Solution:
1. Identify the terms of the sequence:
The given values in the arithmetic sequence are:
[tex]\(a_1 = 9\)[/tex], [tex]\(a_2 = 3\)[/tex], [tex]\(a_3 = -3\)[/tex].
2. Determine the common difference:
The common difference [tex]\(d\)[/tex] in an arithmetic sequence is found by subtracting any term from the following term.
- [tex]\(d = a_2 - a_1 = 3 - 9 = -6\)[/tex]
- We can verify the common difference using the next term:
[tex]\(d = a_3 - a_2 = -3 - 3 = -6\)[/tex]
3. Find the first term:
The first term of the sequence is given directly from the table:
[tex]\[a_1 = 9\][/tex]
4. Formulate the explicit formula:
The general formula for the [tex]\(n\)[/tex]th term in an arithmetic sequence can be expressed as:
[tex]\[a_n = a_1 + (n - 1) \cdot d\][/tex]
Substituting the values we have:
[tex]\[a_n = 9 + (n - 1) \cdot (-6)\][/tex]
Simplifying this, we get:
[tex]\[a_n = 9 - 6(n - 1)\][/tex]
5. Simplify the explicit formula:
Further simplification yields:
[tex]\[a_n = 9 - 6n + 6\][/tex]
[tex]\[a_n = 15 - 6n\][/tex]
6. Determine domain restrictions:
Since the values of [tex]\(n\)[/tex] given in the table start from 1 and increase, the domain of [tex]\(n\)[/tex] in this particular problem is:
[tex]\[n \geq 1\][/tex]
Putting it all together, the explicit formula for the arithmetic sequence is:
[tex]\[a_n = 15 - 6n\][/tex]
with the restriction:
[tex]\[n \geq 1\][/tex]
Therefore, the correct option is:
[tex]\[a_n = 9 - 6(n - 1) \text{ where } n \geq 1\][/tex]
