To determine if a sequence is an arithmetic sequence, we need to check if the difference between consecutive terms is constant. This difference is known as the "common difference." Let’s examine each sequence step-by-step:
### Sequence 1: [tex]\(3, -1, -5, -9, \ldots\)[/tex]
1. Calculate the differences between consecutive terms:
[tex]\[
-1 - 3 = -4
\][/tex]
[tex]\[
-5 - (-1) = -4
\][/tex]
[tex]\[
-9 - (-5) = -4
\][/tex]
The common difference is [tex]\(-4\)[/tex] for all consecutive terms. Since the difference is constant, the sequence is an arithmetic sequence.
### Sequence 2: [tex]\(-2, 0, 4, 8, \ldots\)[/tex]
1. Calculate the differences between consecutive terms:
[tex]\[
0 - (-2) = 2
\][/tex]
[tex]\[
4 - 0 = 4
\][/tex]
[tex]\[
8 - 4 = 4
\][/tex]
Here, the differences are not consistent. The differences are [tex]\(2, 4,\)[/tex] and [tex]\(4\)[/tex]. Since the difference isn’t constant, the sequence is not an arithmetic sequence.
### Sequence 3: [tex]\(1, 3, 9, 27, \ldots\)[/tex]
1. Calculate the differences between consecutive terms:
[tex]\[
3 - 1 = 2
\][/tex]
[tex]\[
9 - 3 = 6
\][/tex]
[tex]\[
27 - 9 = 18
\][/tex]
The differences are [tex]\(2, 6,\)[/tex] and [tex]\(18\)[/tex], which are not constant. Hence, the sequence is not an arithmetic sequence.
### Sequence 4: [tex]\(1, 2, 4, 7, \ldots\)[/tex]
1. Calculate the differences between consecutive terms:
[tex]\[
2 - 1 = 1
\][/tex]
[tex]\[
4 - 2 = 2
\][/tex]
[tex]\[
7 - 4 = 3
\][/tex]
The differences are [tex]\(1, 2,\)[/tex] and [tex]\(3\)[/tex], which are not constant. Therefore, the sequence is not an arithmetic sequence.
### Conclusion
After evaluating each sequence:
- [tex]\(3, -1, -5, -9, \ldots\)[/tex] is an arithmetic sequence.
- [tex]\(-2, 0, 4, 8, \ldots\)[/tex] is not an arithmetic sequence.
- [tex]\(1, 3, 9, 27, \ldots\)[/tex] is not an arithmetic sequence.
- [tex]\(1, 2, 4, 7, \ldots\)[/tex] is not an arithmetic sequence.
The arithmetic sequence among the given options is:
[tex]\[
3, -1, -5, -9, \ldots
\][/tex]
