To determine which of the given sequences are arithmetic, let's review the definition of an arithmetic sequence:
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference.
Let's analyze each sequence step by step:
### Sequence A: [tex]\(2, 6, 18, 54, \ldots\)[/tex]
1. Calculate the differences between consecutive terms:
- [tex]\(6 - 2 = 4\)[/tex]
- [tex]\(18 - 6 = 12\)[/tex]
- [tex]\(54 - 18 = 36\)[/tex]
2. The differences are [tex]\(4, 12, 36\)[/tex] and they are not constant.
Since the differences are not consistent, sequence [tex]\(A\)[/tex] is not an arithmetic sequence.
### Sequence B: [tex]\(2, 2, 2, 2, 2, \ldots\)[/tex]
1. Calculate the differences between consecutive terms:
- [tex]\(2 - 2 = 0\)[/tex]
- [tex]\(2 - 2 = 0\)[/tex]
- [tex]\(2 - 2 = 0\)[/tex]
- [tex]\(2 - 2 = 0\)[/tex]
2. All differences are [tex]\(0\)[/tex], which is consistent.
Since the difference between all consecutive terms is constant, sequence [tex]\(B\)[/tex] is an arithmetic sequence.
### Sequence C: [tex]\(100, 230, 360, 490, \ldots\)[/tex]
1. Calculate the differences between consecutive terms:
- [tex]\(230 - 100 = 130\)[/tex]
- [tex]\(360 - 230 = 130\)[/tex]
- [tex]\(490 - 360 = 130\)[/tex]
2. The differences are [tex]\(130, 130, 130\)[/tex] and they are constant.
Since the difference between all consecutive terms is constant, sequence [tex]\(C\)[/tex] is an arithmetic sequence.
### Sequence D: [tex]\(1, 2, 3, 4, 5, 6, \ldots\)[/tex]
1. Calculate the differences between consecutive terms:
- [tex]\(2 - 1 = 1\)[/tex]
- [tex]\(3 - 2 = 1\)[/tex]
- [tex]\(4 - 3 = 1\)[/tex]
- [tex]\(5 - 4 = 1\)[/tex]
- [tex]\(6 - 5 = 1\)[/tex]
2. The differences are [tex]\(1, 1, 1, 1, 1\)[/tex] and they are constant.
Since the difference between all consecutive terms is constant, sequence [tex]\(D\)[/tex] is an arithmetic sequence.
### Conclusion
After analyzing each sequence, we can conclude that the following sequences are arithmetic:
- Sequence B: [tex]\(2, 2, 2, 2, 2, \ldots\)[/tex]
- Sequence C: [tex]\(100, 230, 360, 490, \ldots\)[/tex]
- Sequence D: [tex]\(1, 2, 3, 4, 5, 6, \ldots\)[/tex]
Sequence [tex]\(A\)[/tex] is not an arithmetic sequence. Therefore, the arithmetic sequences are:
- Sequence B
- Sequence C
- Sequence D
