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Sort the sequences according to whether they are arithmetic, geometric, or neither.

1. [tex]98.3, 94.1, 89.9, 85.7, \ldots[/tex]
2. [tex]1, 0, -1, 0, \ldots[/tex]
3. [tex]1.75, 3.5, 7, 14[/tex]
4. [tex]-12, -10.8, -9.6, -8.4[/tex]
5. [tex]-1, 1, -1, 1, \ldots[/tex]

Answer :

To determine whether the sequences are arithmetic, geometric, or neither, let's analyze each sequence separately:

1. Sequence: 98.3, 94.1, 89.9, 85.7, ...

- Arithmetic Check:
- Calculate the difference between consecutive terms:
- 94.1 - 98.3 = -4.2
- 89.9 - 94.1 = -4.2
- 85.7 - 89.9 = -4.2
- The common difference is -4.2. Since the difference between consecutive terms is constant, this is an arithmetic sequence.

- Conclusion: Arithmetic

2. Sequence: 1, 0, -1, 0, ...

- Arithmetic Check:
- Calculate the difference between consecutive terms:
- 0 - 1 = -1
- -1 - 0 = -1
- 0 - (-1) = 1
- The differences are not consistent.

- Geometric Check:
- Calculate the ratio between consecutive terms:
- 0/1 = 0 (Note: Ratio cannot include division by zero)
- (-1)/0 = undefined
- The ratios are undefined or inconsistent.

- Conclusion: Neither

3. Sequence: 1.75, 3.5, 7, 14, ...

- Geometric Check:
- Calculate the ratio between consecutive terms:
- 3.5 / 1.75 = 2
- 7 / 3.5 = 2
- 14 / 7 = 2
- The common ratio is 2. The ratio between consecutive terms is constant.

- Conclusion: Geometric

4. Sequence: -12, -10.8, -9.6, -8.4, ...

- Arithmetic Check:
- Calculate the difference between consecutive terms:
- -10.8 - (-12) = 1.2
- -9.6 - (-10.8) = 1.2
- -8.4 - (-9.6) = 1.2
- The common difference is 1.2. The difference between consecutive terms is constant.

- Conclusion: Arithmetic

5. Sequence: -1, 1, -1, 1, ...

- Arithmetic Check:
- The differences alternate and are not constant.

- Geometric Check:
- Calculate the ratio between consecutive terms:
- 1 / (-1) = -1
- (-1) / 1 = -1
- 1 / (-1) = -1
- The ratio alternates between -1 and 1, indicating a repeating pattern.

- Conclusion: Neither

Summary:

- The sequence 98.3, 94.1, 89.9, 85.7 is arithmetic.
- The sequence 1, 0, -1, 0 is neither.
- The sequence 1.75, 3.5, 7, 14 is geometric.
- The sequence -12, -10.8, -9.6, -8.4 is arithmetic.
- The sequence -1, 1, -1, 1 is neither.

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