Answer :
Let's thoroughly analyze each sequence to determine which one is an arithmetic sequence. An arithmetic sequence is one in which the difference between consecutive terms is constant.
Sequence A: [tex]\(1, 1, 2, 3, 5, 8, \ldots\)[/tex]
To check if this sequence is arithmetic, let's compute the differences between consecutive terms:
- [tex]\(1 - 1 = 0\)[/tex]
- [tex]\(2 - 1 = 1\)[/tex]
- [tex]\(3 - 2 = 1\)[/tex]
- [tex]\(5 - 3 = 2\)[/tex]
- [tex]\(8 - 5 = 3\)[/tex]
The differences are [tex]\(0, 1, 1, 2, 3\)[/tex], which are not constant. Therefore, Sequence A is not an arithmetic sequence.
Sequence B: [tex]\(-2, 3, 8, 13, 18, \ldots\)[/tex]
Now let's compute the differences between consecutive terms for this sequence:
- [tex]\(3 - (-2) = 5\)[/tex]
- [tex]\(8 - 3 = 5\)[/tex]
- [tex]\(13 - 8 = 5\)[/tex]
- [tex]\(18 - 13 = 5\)[/tex]
The differences are [tex]\(5, 5, 5, 5\)[/tex], which are constant. Therefore, Sequence B is an arithmetic sequence.
Sequence C: [tex]\(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots\)[/tex]
Let's compute the differences between consecutive terms:
- [tex]\(\frac{1}{2} - 1 = -\frac{1}{2}\)[/tex]
- [tex]\(\frac{1}{4} - \frac{1}{2} = -\frac{1}{4}\)[/tex]
- [tex]\(\frac{1}{8} - \frac{1}{4} = -\frac{1}{8}\)[/tex]
The differences are [tex]\(-\frac{1}{2}, -\frac{1}{4}, -\frac{1}{8}\)[/tex], which are not constant. Therefore, Sequence C is not an arithmetic sequence.
Sequence D: [tex]\(4, -4, 4, -4, 4, \ldots\)[/tex]
Let's compute the differences between consecutive terms:
- [tex]\(-4 - 4 = -8\)[/tex]
- [tex]\(4 - (-4) = 8\)[/tex]
- [tex]\(-4 - 4 = -8\)[/tex]
- [tex]\(4 - (-4) = 8\)[/tex]
The differences are [tex]\(-8, 8, -8, 8\)[/tex], which are not constant. Therefore, Sequence D is not an arithmetic sequence.
Given the detailed analysis, the only arithmetic sequence is:
B. [tex]\(-2, 3, 8, 13, 18, \ldots\)[/tex]
Thus, the correct choice is [tex]\( \boxed{2} \)[/tex].
Sequence A: [tex]\(1, 1, 2, 3, 5, 8, \ldots\)[/tex]
To check if this sequence is arithmetic, let's compute the differences between consecutive terms:
- [tex]\(1 - 1 = 0\)[/tex]
- [tex]\(2 - 1 = 1\)[/tex]
- [tex]\(3 - 2 = 1\)[/tex]
- [tex]\(5 - 3 = 2\)[/tex]
- [tex]\(8 - 5 = 3\)[/tex]
The differences are [tex]\(0, 1, 1, 2, 3\)[/tex], which are not constant. Therefore, Sequence A is not an arithmetic sequence.
Sequence B: [tex]\(-2, 3, 8, 13, 18, \ldots\)[/tex]
Now let's compute the differences between consecutive terms for this sequence:
- [tex]\(3 - (-2) = 5\)[/tex]
- [tex]\(8 - 3 = 5\)[/tex]
- [tex]\(13 - 8 = 5\)[/tex]
- [tex]\(18 - 13 = 5\)[/tex]
The differences are [tex]\(5, 5, 5, 5\)[/tex], which are constant. Therefore, Sequence B is an arithmetic sequence.
Sequence C: [tex]\(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots\)[/tex]
Let's compute the differences between consecutive terms:
- [tex]\(\frac{1}{2} - 1 = -\frac{1}{2}\)[/tex]
- [tex]\(\frac{1}{4} - \frac{1}{2} = -\frac{1}{4}\)[/tex]
- [tex]\(\frac{1}{8} - \frac{1}{4} = -\frac{1}{8}\)[/tex]
The differences are [tex]\(-\frac{1}{2}, -\frac{1}{4}, -\frac{1}{8}\)[/tex], which are not constant. Therefore, Sequence C is not an arithmetic sequence.
Sequence D: [tex]\(4, -4, 4, -4, 4, \ldots\)[/tex]
Let's compute the differences between consecutive terms:
- [tex]\(-4 - 4 = -8\)[/tex]
- [tex]\(4 - (-4) = 8\)[/tex]
- [tex]\(-4 - 4 = -8\)[/tex]
- [tex]\(4 - (-4) = 8\)[/tex]
The differences are [tex]\(-8, 8, -8, 8\)[/tex], which are not constant. Therefore, Sequence D is not an arithmetic sequence.
Given the detailed analysis, the only arithmetic sequence is:
B. [tex]\(-2, 3, 8, 13, 18, \ldots\)[/tex]
Thus, the correct choice is [tex]\( \boxed{2} \)[/tex].
