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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]-12, -10.8, -9.6, -8.4, \ldots[/tex]

4. [tex]-1, 1, -1, 1, \ldots[/tex]

Answer :

Sure! Let's take a look at each sequence and determine if they are arithmetic, geometric, or neither.

1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]

- Arithmetic Sequence: This kind of sequence has a common difference between each term. To check:
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]

All differences are the same, so this sequence is arithmetic.

2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]

- Arithmetic Sequence: The differences between consecutive terms are:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]

The differences are not consistent, so it's not arithmetic.

- Geometric Sequence: The ratios between consecutive terms are not defined for division by zero. Thus, this sequence is neither.

3. Sequence 3: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]

- Arithmetic Sequence: Check for common differences:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]

These differences are consistent, indicating that this sequence is arithmetic.

4. Sequence 4: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]

- Arithmetic Sequence: Differences:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]

These differences alternate, so it's not arithmetic.

- Geometric Sequence: Ratios:
- [tex]\(\frac{1}{-1} = -1\)[/tex]
- [tex]\(\frac{-1}{1} = -1\)[/tex]
- [tex]\(\frac{1}{-1} = -1\)[/tex]

The common ratio is consistent, so this sequence is geometric.

Based on these observations:

- Sequence 1 is arithmetic.
- Sequence 2 is neither arithmetic nor geometric.
- Sequence 3 is arithmetic.
- Sequence 4 is geometric.

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