Certainly! Let's solve the problem step by step.
First, let's analyze the given sequence:
[tex]\[ 3, 1, -1, -3, \ldots \][/tex]
This is an arithmetic sequence, as there is a constant difference between each term.
1. Identify the First Term:
The first term of the sequence, denoted by [tex]\(a_1\)[/tex], is:
[tex]\[ a_1 = 3 \][/tex]
2. Find the Common Difference:
The common difference [tex]\(d\)[/tex] is the difference between any two consecutive terms. Let's find [tex]\(d\)[/tex] by subtracting the second term from the first term:
[tex]\[ d = 1 - 3 = -2 \][/tex]
So, the common difference [tex]\(d\)[/tex] is [tex]\(-2\)[/tex].
3. Determine the Term Position:
We need to find the 12th term in the sequence, which can be denoted as [tex]\(a_{12}\)[/tex].
4. Use the General Formula for the [tex]\(n\)[/tex]-th Term:
The formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n-1) \cdot d \][/tex]
In this case, we want to find [tex]\(a_{12}\)[/tex]:
[tex]\[ a_{12} = a_1 + (12-1) \cdot d \][/tex]
5. Substitute the Known Values:
Substitute [tex]\(a_1 = 3\)[/tex], [tex]\(d = -2\)[/tex], and [tex]\(n = 12\)[/tex] into the formula:
[tex]\[ a_{12} = 3 + (12-1) \cdot (-2) \][/tex]
[tex]\[ a_{12} = 3 + 11 \cdot (-2) \][/tex]
[tex]\[ a_{12} = 3 + (-22) \][/tex]
[tex]\[ a_{12} = 3 - 22 \][/tex]
[tex]\[ a_{12} = -19 \][/tex]
So, the 12th term of the sequence, [tex]\(a_{12}\)[/tex], is:
[tex]\[ \boxed{-19} \][/tex]
