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

1. [tex]$98.3, 94.1, 89.9, 85.7$[/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]$-1, 1, -1, 1$[/tex]

Arithmetic:

- [tex]$98.3, 94.1, 89.9, 85.7$[/tex]
- [tex]$-12, -10.8, -9.6, -8.4$[/tex]

Geometric:

- [tex]$1.75, 3.5, 7, 14$[/tex]

Neither:

- [tex]$1, 0, -1, 0, \ldots$[/tex]
- [tex]$-1, 1, -1, 1$[/tex]

Answer :

Certainly! Let's determine whether each sequence is arithmetic, geometric, or neither by analyzing them one by one:

1. Sequence: 98.3, 94.1, 89.9, 85.7
- Arithmetic Check: In an arithmetic sequence, there should be a common difference between consecutive terms. Let's check:
- 94.1 - 98.3 = -4.2
- 89.9 - 94.1 = -4.2
- 85.7 - 89.9 = -4.2
- Since the difference between consecutive terms is constant (-4.2), this sequence is Arithmetic.

2. Sequence: 1, 0, -1, 0
- Arithmetic Check: Check for a common difference:
- 0 - 1 = -1
- -1 - 0 = -1
- 0 - (-1) = 1
- The differences aren't consistent.
- Geometric Check: Check for a common ratio:
- Since a zero term is involved, this sequence can't have a consistent non-zero ratio.
- Neither a common difference nor a common ratio is found, so this sequence is Neither.

3. Sequence: 1.75, 3.5, 7, 14
- Arithmetic Check: Check for a common difference:
- 3.5 - 1.75 = 1.75
- 7 - 3.5 = 3.5
- 14 - 7 = 7
- The differences aren't constant.
- Geometric Check: Check for a common ratio:
- 3.5 / 1.75 = 2
- 7 / 3.5 = 2
- 14 / 7 = 2
- The ratio between consecutive terms is constant (2), so this sequence is Geometric.

4. Sequence: -12, -10.8, -9.6, -8.4
- Arithmetic Check: Check for a common difference:
- -10.8 - (-12) = 1.2
- -9.6 - (-10.8) = 1.2
- -8.4 - (-9.6) = 1.2
- Since the differences are constant (1.2), this sequence is Arithmetic.

5. Sequence: -1, 1, -1, 1
- Arithmetic Check: Check for a common difference:
- 1 - (-1) = 2
- -1 - 1 = -2
- 1 - (-1) = 2
- The differences aren't consistent.
- Geometric Check: Alternating signs make it impossible to have a consistent ratio.
- Without a consistent setup, this sequence is Neither.

In summary:
- First sequence is Arithmetic.
- Second sequence is Neither.
- Third sequence is Geometric.
- Fourth sequence is Arithmetic.
- Fifth sequence is Neither.

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