To find the explicit equation and domain for the given arithmetic sequence, let's go through the problem step by step.
1. Identify the Terms:
The first term ([tex]\(a_1\)[/tex]) is 5.
The second term ([tex]\(a_2\)[/tex]) is 3.
2. Find the Common Difference:
The common difference ([tex]\(d\)[/tex]) in an arithmetic sequence can be calculated by subtracting the first term from the second term:
[tex]\[
d = a_2 - a_1 = 3 - 5 = -2
\][/tex]
3. Formulate the Explicit Equation:
The general formula for the [tex]\(n\)[/tex]th term ([tex]\(a_n\)[/tex]) of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
Substituting the known values ([tex]\(a_1 = 5\)[/tex] and [tex]\(d = -2\)[/tex]) into the formula, we get:
[tex]\[
a_n = 5 + (n-1) \cdot (-2)
\][/tex]
Simplifying this equation:
[tex]\[
a_n = 5 - 2(n - 1)
\][/tex]
So, the explicit equation is:
[tex]\[
a_n = 5 - 2(n - 1)
\][/tex]
4. Determine the Domain:
The domain of [tex]\(n\)[/tex] in an arithmetic sequence is based on the fact that [tex]\(n\)[/tex] represents the position of the term in the sequence. Since the sequence starts at the first term, [tex]\(n\)[/tex] must be a positive integer. Therefore, the domain is:
[tex]\[
\text{all integers where } n \geq 1
\][/tex]
5. Answer Verification:
Comparing the given options with our derived values, the explicit equation [tex]\(a_n = 5 - 2(n - 1)\)[/tex] matches with the option [tex]\(a_n=5-2(n-1) ;\)[/tex] and the correct domain is all integers where [tex]\(n \geq 1\)[/tex].
Thus, the correct answer is:
[tex]\[
a_n=5-2(n-1) ; \text{all integers where } n \geq 1
\][/tex]
