To find the 31st term of an arithmetic sequence where the first term [tex]\( a_1 = 26 \)[/tex] and the 22nd term [tex]\( a_{22} = -226 \)[/tex], we need to follow these steps:
1. Identify the formula for the nth term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
where:
- [tex]\( a_n \)[/tex] is the nth term,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( d \)[/tex] is the common difference,
- [tex]\( n \)[/tex] is the term number.
2. Use the information given to find the common difference [tex]\( d \)[/tex]:
Since we know the 22nd term ([tex]\( a_{22} = -226 \)[/tex]):
[tex]\[
a_{22} = a_1 + 21d
\][/tex]
Substituting the given values:
[tex]\[
-226 = 26 + 21d
\][/tex]
Solving for [tex]\( d \)[/tex]:
[tex]\[
-226 - 26 = 21d
\][/tex]
[tex]\[
-252 = 21d
\][/tex]
[tex]\[
d = \frac{-252}{21}
\][/tex]
[tex]\[
d = -12
\][/tex]
3. Use the common difference to find the 31st term:
Now that we have [tex]\( d = -12 \)[/tex] and we need the 31st term ([tex]\( n = 31 \)[/tex]):
[tex]\[
a_{31} = a_1 + 30d
\][/tex]
Substituting the known values:
[tex]\[
a_{31} = 26 + 30(-12)
\][/tex]
[tex]\[
a_{31} = 26 - 360
\][/tex]
[tex]\[
a_{31} = -334
\][/tex]
Therefore, the 31st term of the arithmetic sequence is [tex]\( -334 \)[/tex].
The correct answer is [tex]\( \boxed{-334} \)[/tex].
