Let's solve the problem step by step for each sequence given.
### Sequence 1: [tex]\(1, -1, 1, -1\)[/tex]
To determine if the sequence is arithmetic, we look for a common difference between consecutive terms:
- The difference between the first and second term: [tex]\(-1 - 1 = -2\)[/tex]
- The difference between the second and third term: [tex]\(1 - (-1) = 2\)[/tex]
- The difference between the third and fourth term: [tex]\(-1 - 1 = -2\)[/tex]
Since the differences are not the same, this sequence is not arithmetic.
### Sequence 2: [tex]\(1, -2, -1, -2, \ldots\)[/tex]
Next, let's check the differences between consecutive terms for sequence 2:
- The difference between the first and second term: [tex]\(-2 - 1 = -3\)[/tex]
- The difference between the second and third term: [tex]\(-1 - (-2) = 1\)[/tex]
- The difference between the third and fourth term: [tex]\(-2 - (-1) = -1\)[/tex]
Since the differences are not the same, this sequence is not arithmetic.
### Sequence 3: [tex]\(-3, -6, -9, -12, \ldots\)[/tex]
Now, let’s check the differences between consecutive terms for sequence 3:
- The difference between the first and second term: [tex]\(-6 - (-3) = -3\)[/tex]
- The difference between the second and third term: [tex]\(-9 - (-6) = -3\)[/tex]
- The difference between the third and fourth term: [tex]\(-12 - (-9) = -3\)[/tex]
The differences are consistent, so this sequence is arithmetic with a common difference of [tex]\(-3\)[/tex].
### Sequence 4: [tex]\(2, 4, 6, 8, 10\)[/tex]
Finally, let’s check the differences between consecutive terms for sequence 4:
- The difference between the first and second term: [tex]\(4 - 2 = 2\)[/tex]
- The difference between the second and third term: [tex]\(6 - 4 = 2\)[/tex]
- The difference between the third and fourth term: [tex]\(8 - 6 = 2\)[/tex]
- The difference between the fourth and fifth term: [tex]\(10 - 8 = 2\)[/tex]
The differences are consistent, so this sequence is arithmetic with a common difference of [tex]\(2\)[/tex].
### Conclusion:
1. The sequence [tex]\(1, -1, 1, -1\)[/tex] is not arithmetic.
2. The sequence [tex]\(1, -2, -1, -2, \ldots\)[/tex] is not arithmetic.
3. The sequence [tex]\(-3, -6, -9, -12, \ldots\)[/tex] is arithmetic with a common difference of [tex]\(-3\)[/tex].
4. The sequence [tex]\(2, 4, 6, 8, 10\)[/tex] is arithmetic with a common difference of [tex]\(2\)[/tex].
