Let's analyze the given arithmetic sequence: [tex]\(-27, -36, -45, -54, \ldots\)[/tex]
1. Identify the first term ([tex]\(a_1\)[/tex]) and the common difference ([tex]\(d\)[/tex]).
- The first term, [tex]\(a_1\)[/tex], is clearly [tex]\(-27\)[/tex].
- To find the common difference, [tex]\(d\)[/tex], subtract the first term from the second term:
[tex]\[
d = -36 - (-27) = -36 + 27 = -9
\][/tex]
2. General formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence:
The [tex]\(n\)[/tex]th term of an arithmetic sequence is given by:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
Where:
- [tex]\(a_1\)[/tex] is the first term.
- [tex]\(d\)[/tex] is the common difference.
- [tex]\(n\)[/tex] is the term number.
3. Substitute the values of [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex] into the formula:
[tex]\[
a_n = -27 + (n-1) \cdot (-9)
\][/tex]
4. Simplify the expression:
[tex]\[
a_n = -27 + (n-1)(-9)
\][/tex]
Distribute the [tex]\(-9\)[/tex]:
[tex]\[
a_n = -27 - 9(n-1)
\][/tex]
Simplify inside the parentheses:
[tex]\[
a_n = -27 - 9n + 9
\][/tex]
Combine like terms:
[tex]\[
a_n = -9n - 18
\][/tex]
5. Final formula:
The formula for the [tex]\(n\)[/tex]th term of the arithmetic sequence is:
[tex]\[
a_n = -9n - 18
\][/tex]
6. Match with the given options:
The correct option is:
[tex]\[
\boxed{D. \, a_n = -9n - 18}
\][/tex]
