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Answer :
Sure! Let's go through the sequences one by one to determine whether they are arithmetic, geometric, or neither.
### Sequence Analysis
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- An arithmetic sequence has a constant 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 sequence is Arithmetic.
2. Sequence 2: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- To be an arithmetic sequence, the difference between consecutive terms must be constant.
- Calculate the differences:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- The differences alternate between 2 and -2, hence it's not constant.
- To be a geometric sequence, the ratio between consecutive terms must be constant.
- Calculate the ratios:
- [tex]\(1 / -1 = -1\)[/tex]
- [tex]\(-1 / 1 = -1\)[/tex]
- [tex]\(1 / -1 = -1\)[/tex]
- The ratio is consistent ([tex]\(-1\)[/tex]), hence this sequence is Geometric.
3. Sequence 3: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- An arithmetic sequence checks for a constant difference:
- [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 sequence is Arithmetic.
4. Sequence 4: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- For a geometric sequence, the ratio between consecutive terms must be constant.
- Calculate the ratios:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- The ratio is consistent (2), so this sequence is Geometric.
### Conclusion
- Arithmetic: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Geometric: [tex]\(-1, 1, -1, 1, \ldots\)[/tex] and [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Neither: None of the sequences fit into this category.
This detailed analysis helps classify each sequence correctly based on their properties.
### Sequence Analysis
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- An arithmetic sequence has a constant 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 sequence is Arithmetic.
2. Sequence 2: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- To be an arithmetic sequence, the difference between consecutive terms must be constant.
- Calculate the differences:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- The differences alternate between 2 and -2, hence it's not constant.
- To be a geometric sequence, the ratio between consecutive terms must be constant.
- Calculate the ratios:
- [tex]\(1 / -1 = -1\)[/tex]
- [tex]\(-1 / 1 = -1\)[/tex]
- [tex]\(1 / -1 = -1\)[/tex]
- The ratio is consistent ([tex]\(-1\)[/tex]), hence this sequence is Geometric.
3. Sequence 3: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- An arithmetic sequence checks for a constant difference:
- [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 sequence is Arithmetic.
4. Sequence 4: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- For a geometric sequence, the ratio between consecutive terms must be constant.
- Calculate the ratios:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- The ratio is consistent (2), so this sequence is Geometric.
### Conclusion
- Arithmetic: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Geometric: [tex]\(-1, 1, -1, 1, \ldots\)[/tex] and [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Neither: None of the sequences fit into this category.
This detailed analysis helps classify each sequence correctly based on their properties.
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