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Which of the following are arithmetic sequences? Check all that apply.

A. [tex]2, 6, 18, 54, \ldots[/tex]

B. [tex]2, 2, 2, 2, 2, \ldots[/tex]

C. [tex]100, 230, 360, 490, \ldots[/tex]

D. [tex]1, 2, 3, 4, 5, 6, \ldots[/tex]

Sagot :

GinnyAnswer
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
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