Answer :
Certainly! Let's analyze each of the series given and find their common differences step by step:
### Series 1: [tex]\( 2, -4, -10, -16, -22 \)[/tex]
#### Finding the Common Difference:
1. Take the difference between the second term and the first term:
[tex]\[ -4 - 2 = -6 \][/tex]
2. Take the difference between the third term and the second term:
[tex]\[ -10 - (-4) = -6 \][/tex]
3. Take the difference between the fourth term and the third term:
[tex]\[ -16 - (-10) = -6 \][/tex]
4. Take the difference between the fifth term and the fourth term:
[tex]\[ -22 - (-16) = -6 \][/tex]
Thus, the common difference for series 1 is:
[tex]\(\boxed{-6}\)[/tex]
### Series 2: [tex]\( 2, 8, 14, 20, 26 \)[/tex]
#### Finding the Common Difference:
1. Take the difference between the second term and the first term:
[tex]\[ 8 - 2 = 6 \][/tex]
2. Take the difference between the third term and the second term:
[tex]\[ 14 - 8 = 6 \][/tex]
3. Take the difference between the fourth term and the third term:
[tex]\[ 20 - 14 = 6 \][/tex]
4. Take the difference between the fifth term and the fourth term:
[tex]\[ 26 - 20 = 6 \][/tex]
Thus, the common difference for series 2 is:
[tex]\(\boxed{6}\)[/tex]
### Series 3: [tex]\( -4, -10, -16, -22, -28 \)[/tex]
#### Finding the Common Difference:
1. Take the difference between the second term and the first term:
[tex]\[ -10 - (-4) = -6 \][/tex]
2. Take the difference between the third term and the second term:
[tex]\[ -16 - (-10) = -6 \][/tex]
3. Take the difference between the fourth term and the third term:
[tex]\[ -22 - (-16) = -6 \][/tex]
4. Take the difference between the fifth term and the fourth term:
[tex]\[ -28 - (-22) = -6 \][/tex]
Thus, the common difference for series 3 is:
[tex]\(\boxed{-6}\)[/tex]
### Series 4: [tex]\( -22, -16, -10, -4, 0 \)[/tex]
#### Finding the Common Difference:
1. Take the difference between the second term and the first term:
[tex]\[ -16 - (-22) = 6 \][/tex]
2. Take the difference between the third term and the second term:
[tex]\[ -10 - (-16) = 6 \][/tex]
3. Take the difference between the fourth term and the third term:
[tex]\[ -4 - (-10) = 6 \][/tex]
4. Take the difference between the fifth term and the fourth term:
[tex]\[ 0 - (-4) = 6 \][/tex]
Thus, the common difference for series 4 is:
[tex]\(\boxed{6}\)[/tex]
### Recap of Common Differences for Each Series:
1. The common difference for the first series [tex]\(2, -4, -10, -16, -22\)[/tex] is [tex]\(-6\)[/tex].
2. The common difference for the second series [tex]\(2, 8, 14, 20, 26\)[/tex] is [tex]\(6\)[/tex].
3. The common difference for the third series [tex]\(-4, -10, -16, -22, -28\)[/tex] is [tex]\(-6\)[/tex].
4. The common difference for the fourth series [tex]\(-22, -16, -10, -4, 0\)[/tex] is [tex]\(6\)[/tex].
So the final common differences are:
[tex]\[ (-6, 6, -6, 6) \][/tex]
### Series 1: [tex]\( 2, -4, -10, -16, -22 \)[/tex]
#### Finding the Common Difference:
1. Take the difference between the second term and the first term:
[tex]\[ -4 - 2 = -6 \][/tex]
2. Take the difference between the third term and the second term:
[tex]\[ -10 - (-4) = -6 \][/tex]
3. Take the difference between the fourth term and the third term:
[tex]\[ -16 - (-10) = -6 \][/tex]
4. Take the difference between the fifth term and the fourth term:
[tex]\[ -22 - (-16) = -6 \][/tex]
Thus, the common difference for series 1 is:
[tex]\(\boxed{-6}\)[/tex]
### Series 2: [tex]\( 2, 8, 14, 20, 26 \)[/tex]
#### Finding the Common Difference:
1. Take the difference between the second term and the first term:
[tex]\[ 8 - 2 = 6 \][/tex]
2. Take the difference between the third term and the second term:
[tex]\[ 14 - 8 = 6 \][/tex]
3. Take the difference between the fourth term and the third term:
[tex]\[ 20 - 14 = 6 \][/tex]
4. Take the difference between the fifth term and the fourth term:
[tex]\[ 26 - 20 = 6 \][/tex]
Thus, the common difference for series 2 is:
[tex]\(\boxed{6}\)[/tex]
### Series 3: [tex]\( -4, -10, -16, -22, -28 \)[/tex]
#### Finding the Common Difference:
1. Take the difference between the second term and the first term:
[tex]\[ -10 - (-4) = -6 \][/tex]
2. Take the difference between the third term and the second term:
[tex]\[ -16 - (-10) = -6 \][/tex]
3. Take the difference between the fourth term and the third term:
[tex]\[ -22 - (-16) = -6 \][/tex]
4. Take the difference between the fifth term and the fourth term:
[tex]\[ -28 - (-22) = -6 \][/tex]
Thus, the common difference for series 3 is:
[tex]\(\boxed{-6}\)[/tex]
### Series 4: [tex]\( -22, -16, -10, -4, 0 \)[/tex]
#### Finding the Common Difference:
1. Take the difference between the second term and the first term:
[tex]\[ -16 - (-22) = 6 \][/tex]
2. Take the difference between the third term and the second term:
[tex]\[ -10 - (-16) = 6 \][/tex]
3. Take the difference between the fourth term and the third term:
[tex]\[ -4 - (-10) = 6 \][/tex]
4. Take the difference between the fifth term and the fourth term:
[tex]\[ 0 - (-4) = 6 \][/tex]
Thus, the common difference for series 4 is:
[tex]\(\boxed{6}\)[/tex]
### Recap of Common Differences for Each Series:
1. The common difference for the first series [tex]\(2, -4, -10, -16, -22\)[/tex] is [tex]\(-6\)[/tex].
2. The common difference for the second series [tex]\(2, 8, 14, 20, 26\)[/tex] is [tex]\(6\)[/tex].
3. The common difference for the third series [tex]\(-4, -10, -16, -22, -28\)[/tex] is [tex]\(-6\)[/tex].
4. The common difference for the fourth series [tex]\(-22, -16, -10, -4, 0\)[/tex] is [tex]\(6\)[/tex].
So the final common differences are:
[tex]\[ (-6, 6, -6, 6) \][/tex]
