To solve this problem, let's assume the sequence is an arithmetic sequence. An arithmetic sequence is one in which the difference between consecutive terms (called the common difference) is constant.
Given:
1. First term ([tex]\(a_1\)[/tex]) [tex]\(= 22\)[/tex]
2. Third term ([tex]\(a_3\)[/tex]) [tex]\(= 70\)[/tex]
In an arithmetic sequence, the [tex]\(n\)[/tex]-th term is given by:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
where [tex]\(d\)[/tex] is the common difference.
We need to find the common difference [tex]\(d\)[/tex] and the second term [tex]\(a_2\)[/tex].
### Step 1: Write the expression for the third term
The third term ([tex]\(a_3\)[/tex]) can be expressed as:
[tex]\[ a_3 = a_1 + 2d \][/tex]
### Step 2: Substitute the given values
Given [tex]\(a_3 = 70\)[/tex] and [tex]\(a_1 = 22\)[/tex], we substitute these values into the equation:
[tex]\[ 70 = 22 + 2d \][/tex]
### Step 3: Solve for the common difference [tex]\(d\)[/tex]
First, isolate [tex]\(2d\)[/tex]:
[tex]\[ 70 - 22 = 2d \][/tex]
[tex]\[ 48 = 2d \][/tex]
Now, solve for [tex]\(d\)[/tex]:
[tex]\[ d = \frac{48}{2} \][/tex]
[tex]\[ d = 24 \][/tex]
So, the common difference [tex]\(d\)[/tex] is 24.
### Step 4: Find the second term ([tex]\(a_2\)[/tex])
The second term ([tex]\(a_2\)[/tex]) is given by:
[tex]\[ a_2 = a_1 + d \][/tex]
Substitute the known values:
[tex]\[ a_2 = 22 + 24 \][/tex]
[tex]\[ a_2 = 46 \][/tex]
So, the second term [tex]\(a_2\)[/tex] is 46.
### Conclusion
The common difference [tex]\(d\)[/tex] is 24, and the sequence including the missing number is:
[tex]\[ 22, 46, 70 \][/tex]
