To find the [tex]\(n\)[/tex]th term of an arithmetic sequence, we can use the formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
where:
- [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]th term of the sequence,
- [tex]\(a_1\)[/tex] is the first term of the sequence,
- [tex]\(n\)[/tex] is the position of the term in the sequence,
- [tex]\(d\)[/tex] is the common difference between consecutive terms.
Given:
- The first term [tex]\(a_1 = -12\)[/tex],
- The common difference [tex]\(d = 4\)[/tex].
Now we'll substitute these values into the formula:
1. Write down the general formula for the [tex]\(n\)[/tex]th term:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
2. Substitute [tex]\(a_1 = -12\)[/tex] and [tex]\(d = 4\)[/tex] into the formula:
[tex]\[
a_n = -12 + (n - 1) \cdot 4
\][/tex]
3. Simplify the expression inside the parentheses:
[tex]\[
a_n = -12 + 4n - 4
\][/tex]
4. Combine like terms:
[tex]\[
a_n = 4n - 16
\][/tex]
So, the formula for the [tex]\(n\)[/tex]th term of the arithmetic sequence is:
[tex]\[ a_n = 4n - 16 \][/tex]
In the form [tex]\(a_n = mn + b\)[/tex], we have:
[tex]\[ a_n = 4n - 16 \][/tex]
Therefore, our final answer is:
[tex]\[ a_n = 4n - 16 \][/tex]
