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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 :

Sure, let's determine whether each of the sequences is arithmetic, geometric, or neither by analyzing the patterns in their terms.

1. Sequence 1: 98.3, 94.1, 89.9, 85.7, …
- To check if it's an arithmetic sequence, we look for a common difference between consecutive terms.
- Calculate 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 difference is constant, this is an arithmetic sequence.

2. Sequence 2: 1, 0, -1, 0, …
- Check for arithmetic progression:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- The differences are not constant.
- Check for geometric progression:
- The ratio between terms is not constant as well.
- Thus, this sequence is neither arithmetic nor geometric.

3. Sequence 3: 1.75, 3.5, 7, 14, …
- To check if it's a geometric sequence, we find the common ratio between consecutive terms.
- Calculate the ratios:
- [tex]\(3.5 \div 1.75 = 2\)[/tex]
- [tex]\(7 \div 3.5 = 2\)[/tex]
- [tex]\(14 \div 7 = 2\)[/tex]
- The ratio is constant, making this a geometric sequence.

4. Sequence 4: -12, -10.8, -9.6, -8.4, …
- Look for a common difference to determine if it's arithmetic:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- The difference is constant, so this is an arithmetic sequence.

5. Sequence 5: -1, 1, -1, 1, …
- Check for arithmetic progression:
- The differences alternate between 2 and -2, which are not constant.
- Check for geometric progression:
- The ratios also alternate, not forming a constant ratio.
- Therefore, this sequence is neither arithmetic nor geometric.

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

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