To identify the 31st term of an arithmetic sequence where the first term is [tex]\( a_1 = 26 \)[/tex] and the 22nd term is [tex]\( a_{22} = -226 \)[/tex], we can follow these steps:
1. Determine the Common Difference ([tex]\( d \)[/tex]):
The general formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
For the 22nd term ([tex]\( n = 22 \)[/tex]):
[tex]\[
a_{22} = a_1 + (22 - 1) \cdot d
\][/tex]
Plugging in the given values:
[tex]\[
-226 = 26 + 21 \cdot d
\][/tex]
Solving for [tex]\( d \)[/tex]:
[tex]\[
-226 - 26 = 21 \cdot d
\][/tex]
[tex]\[
-252 = 21 \cdot d
\][/tex]
[tex]\[
d = \frac{-252}{21}
\][/tex]
[tex]\[
d = -12
\][/tex]
2. Find the 31st Term ([tex]\( a_{31} \)[/tex]):
Using the same formula for the [tex]\( n \)[/tex]-th term:
[tex]\[
a_{31} = a_1 + (31 - 1) \cdot d
\][/tex]
Plugging in the values we have:
[tex]\[
a_{31} = 26 + 30 \cdot (-12)
\][/tex]
Calculate the product:
[tex]\[
30 \cdot (-12) = -360
\][/tex]
Add this to the first term:
[tex]\[
a_{31} = 26 + (-360)
\][/tex]
[tex]\[
a_{31} = 26 - 360
\][/tex]
[tex]\[
a_{31} = -334
\][/tex]
Thus, the 31st term of the arithmetic sequence is [tex]\( \boxed{-334} \)[/tex].
