What are the explicit equation and domain for an arithmetic sequence with a first term of 5 and a second term of 3?

A. [tex]a_n = 5 - 3(n - 1), \text{ all integers where } n \geq 1[/tex]
B. [tex]a_n = 5 - 3(n - 1); \text{ all integers where } n \geq 0[/tex]
C. [tex]a_n = 5 - 2(n - 1); \text{ all integers where } n \geq 0[/tex]
D. [tex]a_n = 5 - 2(n - 1); \text{ all integers where } n \geq 1[/tex]



Answer :

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]