To find the 12th term ([tex]\(a_{12}\)[/tex]) of the given arithmetic sequence [tex]\(3, 1, -1, -3, \ldots\)[/tex], we need to follow a few steps. Here’s a detailed, step-by-step solution:
1. Identify the first term ([tex]\(a_1\)[/tex]) and the common difference ([tex]\(d\)[/tex]):
- The first term of the sequence, [tex]\(a_1\)[/tex], is given as [tex]\(3\)[/tex].
- The common difference ([tex]\(d\)[/tex]) is found by subtracting the first term from the second term: [tex]\(1 - 3 = -2\)[/tex].
2. Use the formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
We want to find the 12th term ([tex]\(a_{12}\)[/tex]), so [tex]\(n = 12\)[/tex].
3. Substitute the values into the formula:
[tex]\[
a_{12} = 3 + (12 - 1) \cdot (-2)
\][/tex]
4. Calculate the expression inside the parentheses first:
[tex]\[
12 - 1 = 11
\][/tex]
5. Multiply this result by the common difference [tex]\(d\)[/tex]:
[tex]\[
11 \cdot (-2) = -22
\][/tex]
6. Finally, add this product to the first term [tex]\(a_1\)[/tex]:
[tex]\[
3 + (-22) = 3 - 22 = -19
\][/tex]
Therefore, the 12th term ([tex]\(a_{12}\)[/tex]) of the sequence is [tex]\(-19\)[/tex].
