To determine which term in the arithmetic sequence [tex]\(8, 5, 2, -1, \ldots\)[/tex] is [tex]\(-25\)[/tex], we need to follow these steps:
1. Identify the first term ([tex]\(a_1\)[/tex]) and the common difference ([tex]\(d\)[/tex]):
- The first term [tex]\(a_1\)[/tex] of the sequence is [tex]\(8\)[/tex].
- The common difference [tex]\(d\)[/tex] is calculated by subtracting the first term from the second term:
[tex]\[
d = 5 - 8 = -3
\][/tex]
2. Use the formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence:
The formula to find the [tex]\(n\)[/tex]th term ([tex]\(a_n\)[/tex]) in an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1) d
\][/tex]
3. Set up the equation to find [tex]\(n\)[/tex] when [tex]\(a_n = -25\)[/tex]:
We want to find the term where [tex]\(a_n = -25\)[/tex]. Substitute the given values into the formula:
[tex]\[
-25 = 8 + (n - 1) (-3)
\][/tex]
4. Solve for [tex]\(n\)[/tex]:
Simplify the equation step-by-step to solve for [tex]\(n\)[/tex]:
[tex]\[
-25 = 8 + (n - 1) (-3)
\][/tex]
[tex]\[
-25 = 8 - 3(n - 1)
\][/tex]
[tex]\[
-25 = 8 - 3n + 3
\][/tex]
[tex]\[
-25 = 11 - 3n
\][/tex]
[tex]\[
-25 - 11 = -3n
\][/tex]
[tex]\[
-36 = -3n
\][/tex]
[tex]\[
n = \frac{-36}{-3}
\][/tex]
[tex]\[
n = 12
\][/tex]
Thus, -25 is the 12th term in the arithmetic sequence.
