To find the formula for the [tex]\(n\)[/tex]th term of the given arithmetic sequence where the first few terms are [tex]\(a_1 = 8\)[/tex], [tex]\(a_2 = 4\)[/tex], [tex]\(a_3 = 0\)[/tex], and [tex]\(a_4 = -4\)[/tex], we need to follow these steps:
1. Identify the Common Difference [tex]\(d\)[/tex]:
The common difference [tex]\(d\)[/tex] in an arithmetic sequence can be found by subtracting any term from the term that follows it.
[tex]\[
d = a_2 - a_1 = 4 - 8 = -4
\][/tex]
2. Determine the General Formula for the [tex]\(n\)[/tex]th Term:
The general formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence is given by:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
Here, [tex]\(a_1\)[/tex] is the first term, and [tex]\(d\)[/tex] is the common difference.
3. Substitute the Known Values:
We know [tex]\(a_1 = 8\)[/tex] and [tex]\(d = -4\)[/tex]. Substitute these values into the general formula.
[tex]\[
a_n = 8 + (n - 1) \cdot (-4)
\][/tex]
4. Simplify the Expression:
Distribute the [tex]\((-4)\)[/tex] through the [tex]\((n - 1)\)[/tex]:
[tex]\[
a_n = 8 + (n - 1) \times (-4)
\][/tex]
[tex]\[
a_n = 8 + (n \cdot -4 - 1 \cdot -4)
\][/tex]
[tex]\[
a_n = 8 + (-4n + 4)
\][/tex]
Combine like terms:
[tex]\[
a_n = 8 + 4 - 4n
\][/tex]
[tex]\[
a_n = 12 - 4n
\][/tex]
Thus, the formula for the [tex]\(n\)[/tex]th term in this arithmetic sequence is:
[tex]\[
a_n = 12 - 4n
\][/tex]
So the final equation in the requested format is:
[tex]\[
a_n = 12 - 4n
\][/tex]
