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Answer :
Let's sort the given sequences according to whether they are arithmetic, geometric, or neither.
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- This sequence has a common difference between consecutive terms.
- The difference between each term is [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]\(1, 0, -1, 0, \ldots\)[/tex]
- This sequence alternates between 1, 0, and -1.
- There is no constant difference, nor a common ratio present.
- Therefore, this sequence is neither arithmetic nor geometric.
3. Sequence 3: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- Calculate the ratio of consecutive terms: [tex]\(3.5 / 1.75 = 2\)[/tex], [tex]\(7 / 3.5 = 2\)[/tex], [tex]\(14 / 7 = 2\)[/tex].
- The ratio is constant, so this sequence is geometric.
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Find the common 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, indicating this sequence is arithmetic.
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- This sequence alternates between -1 and 1.
- There is no constant difference between terms nor a consistent ratio.
- Thus, this sequence is neither arithmetic nor geometric.
In summary, the sequences sorted by type are:
- Arithmetic: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Geometric: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- Neither: [tex]\(1, 0, -1, 0, \ldots\)[/tex] and [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- This sequence has a common difference between consecutive terms.
- The difference between each term is [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]\(1, 0, -1, 0, \ldots\)[/tex]
- This sequence alternates between 1, 0, and -1.
- There is no constant difference, nor a common ratio present.
- Therefore, this sequence is neither arithmetic nor geometric.
3. Sequence 3: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- Calculate the ratio of consecutive terms: [tex]\(3.5 / 1.75 = 2\)[/tex], [tex]\(7 / 3.5 = 2\)[/tex], [tex]\(14 / 7 = 2\)[/tex].
- The ratio is constant, so this sequence is geometric.
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Find the common 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, indicating this sequence is arithmetic.
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- This sequence alternates between -1 and 1.
- There is no constant difference between terms nor a consistent ratio.
- Thus, this sequence is neither arithmetic nor geometric.
In summary, the sequences sorted by type are:
- Arithmetic: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Geometric: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- Neither: [tex]\(1, 0, -1, 0, \ldots\)[/tex] and [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
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