To find the 22nd term of the arithmetic sequence where \( a_1 = 8 \) and \( a_9 = 56 \), we need to follow these steps:
1. Understand the General Formula:
In an arithmetic sequence, the \( n \)-th term \( a_n \) can be found using the formula:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
where \( d \) is the common difference.
2. Identify Known Values:
In this problem, we are given:
- The first term: \( a_1 = 8 \)
- The ninth term: \( a_9 = 56 \)
3. Set Up the Equation for the Ninth Term:
Using the general formula for the ninth term:
[tex]\[
a_9 = a_1 + (9-1) \cdot d
\][/tex]
Substituting the known values:
[tex]\[
56 = 8 + 8d
\][/tex]
4. Solve for the Common Difference \( d \):
Simplify the equation:
[tex]\[
56 = 8 + 8d
\][/tex]
Subtract 8 from both sides:
[tex]\[
48 = 8d
\][/tex]
Divide both sides by 8:
[tex]\[
d = 6
\][/tex]
5. Find the 22nd Term:
Now that we know the common difference \( d \), we can find the 22nd term \( a_{22} \).
Using the general formula for the 22nd term:
[tex]\[
a_{22} = a_1 + (22-1) \cdot d
\][/tex]
Substitute the known values:
[tex]\[
a_{22} = 8 + 21 \cdot 6
\][/tex]
Calculate \( 21 \cdot 6 \):
[tex]\[
21 \cdot 6 = 126
\][/tex]
Add this to the first term:
[tex]\[
a_{22} = 8 + 126 = 134
\][/tex]
Thus, the 22nd term of the arithmetic sequence is [tex]\(\boxed{134}\)[/tex].
