To identify the 27th term of an arithmetic sequence where the first term [tex]\(a_1 = 38\)[/tex] and the 17th term [tex]\(a_{17} = -74\)[/tex], we will follow these steps:
1. Determine the common difference [tex]\(d\)[/tex]:
- The general form of the [tex]\(n\)[/tex]-th term of an arithmetic sequence is given by:
[tex]\[
a_n = a_1 + (n - 1)d
\][/tex]
- For the 17th term ([tex]\(a_{17}\)[/tex]):
[tex]\[
a_{17} = a_1 + (17 - 1)d
\][/tex]
Plugging in the values we know:
[tex]\[
-74 = 38 + 16d
\][/tex]
- Solving for [tex]\(d\)[/tex]:
[tex]\[
-74 - 38 = 16d
\][/tex]
[tex]\[
-112 = 16d
\][/tex]
[tex]\[
d = \frac{-112}{16}
\][/tex]
[tex]\[
d = -7
\][/tex]
2. Find the 27th term [tex]\(a_{27}\)[/tex]:
- Using the general formula for the [tex]\(n\)[/tex]-th term again:
[tex]\[
a_{27} = a_1 + (27 - 1)d
\][/tex]
Plugging in the values we know:
[tex]\[
a_{27} = 38 + 26(-7)
\][/tex]
- Calculating the value:
[tex]\[
a_{27} = 38 - 182
\][/tex]
[tex]\[
a_{27} = -144
\][/tex]
Therefore, the 27th term of the arithmetic sequence is [tex]\(-144\)[/tex].
