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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]

**Options:**

- Arithmetic
- Geometric
- Neither

Answer :

Let's analyze each sequence to determine if it is arithmetic, geometric, or neither.

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

- Arithmetic Check: In an arithmetic sequence, the difference between consecutive terms is constant.
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- The common difference is [tex]\(-4.2\)[/tex], so this is an Arithmetic sequence.

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

- Arithmetic Check:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- The differences are not the same.
- Geometric Check: In a geometric sequence, the ratio between consecutive terms is constant.
- [tex]\(0 / 1 = 0\)[/tex]
- [tex]\(-1 / 0\)[/tex] is undefined.
- The ratio is inconsistent, and division by zero is problematic.
- This sequence is Neither arithmetic nor geometric.

3. Sequence: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]

- Arithmetic Check:
- [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- [tex]\(7 - 3.5 = 3.5\)[/tex]
- Differences are not the same.
- Geometric Check:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- The common ratio is [tex]\(2\)[/tex]; this is a Geometric sequence.

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

- Arithmetic Check:
- [tex]\(-10.8 + 12 = 1.2\)[/tex]
- [tex]\(-9.6 + 10.8 = 1.2\)[/tex]
- [tex]\(-8.4 + 9.6 = 1.2\)[/tex]
- The common difference is [tex]\(1.2\)[/tex], so this is an Arithmetic sequence.

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

- Arithmetic Check:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- Differences are inconsistent.
- Geometric Check:
- [tex]\(1 / (-1) = -1\)[/tex]
- [tex]\(-1 / 1 = -1\)[/tex]
- [tex]\(1 / (-1) = -1\)[/tex]
- The common ratio is [tex]\(-1\)[/tex]; this is a Geometric sequence.

In summary:
- [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] is Arithmetic
- [tex]\(1, 0, -1, 0, \ldots\)[/tex] is Neither
- [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex] is Geometric
- [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex] is Arithmetic
- [tex]\(-1, 1, -1, 1, \ldots\)[/tex] is Geometric

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