High School

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]$1.75, 35, 7, 14$[/tex]
4. [tex]$-12, -10.8, -9.6, -8.4$[/tex]
5. [tex]$-1, 1, -1, 1, \ldots$[/tex]

Answer :

Let's go through each sequence to determine if they are arithmetic, geometric, or neither:

1. Sequence 1: 98.3, 94.1, 89.9, 85.7, ...
- An arithmetic sequence has a constant difference between consecutive terms.
- Check the differences:
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- Since the differences are equal, this is an arithmetic sequence.

2. Sequence 2: 1, 0, -1, 0, ...
- An arithmetic sequence has a constant difference, and a geometric sequence has a constant ratio.
- The sequence alternates between 1, 0, and -1, which does not follow a constant difference or ratio.
- Therefore, this sequence is neither arithmetic nor geometric.

3. Sequence 3: 1.75, 35, 7, 14
- A geometric sequence has a constant ratio between consecutive terms.
- Check the ratios:
- [tex]\(35 / 1.75 = 20\)[/tex]
- [tex]\(7 / 35 = 0.2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- The ratios are not equal, so this is neither a geometric nor an arithmetic sequence.

4. Sequence 4: -12, -10.8, -9.6, -8.4
- An arithmetic sequence has a constant difference between consecutive terms.
- Check the differences:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- Since the differences are equal, this is an arithmetic sequence.

5. Sequence 5: -1, 1, -1, 1, ...
- Similar to sequence 2, it alternates and does not have a constant difference or ratio.
- Therefore, this sequence is neither arithmetic nor geometric.

In summary:
- Sequence 1 is arithmetic.
- Sequence 2 is neither.
- Sequence 3 is neither.
- Sequence 4 is arithmetic.
- Sequence 5 is neither.

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