Answer :

GinnyAnswer
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].

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