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Answer :
Let's analyze each sequence one by one to determine whether they are arithmetic, geometric, or neither.
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7,\ldots\)[/tex]
- Arithmetic Check: In an arithmetic sequence, the difference between consecutive terms is constant. Here, the differences are:
- [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 the same, this sequence is arithmetic.
- Geometric Check: In a geometric sequence, the ratio of consecutive terms must be constant, which is not true in this case.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Arithmetic Check: The differences between terms vary, so it's not arithmetic.
- Geometric Check: Ratios also vary between terms, so it's not geometric.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Arithmetic Check: Differences between terms are not constant, so it's not arithmetic.
- Geometric Check: The ratio of each term to the previous one is constant:
- [tex]\(\frac{3.5}{1.75} = 2\)[/tex]
- [tex]\(\frac{7}{3.5} = 2\)[/tex]
- [tex]\(\frac{14}{7} = 2\)[/tex]
Since the ratios are the same, this sequence is geometric.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Arithmetic Check: 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 constant, this sequence is arithmetic.
- Geometric Check: Ratios are not constant, so it's not geometric.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Arithmetic Check: The differences between terms are not constant, so it's not arithmetic.
- Geometric Check: Ratios are not constant either; it keeps alternating, so it's not geometric.
Now, summarizing the sequences:
- Sequence [tex]\(98.3, 94.1, 89.9, 85.7,\ldots\)[/tex] is Arithmetic.
- Sequence [tex]\(1, 0, -1, 0, \ldots\)[/tex] is Neither.
- Sequence [tex]\(1.75, 3.5, 7, 14\)[/tex] is Geometric.
- Sequence [tex]\(-12, -10.8, -9.6, -8.4\)[/tex] is Arithmetic.
- Sequence [tex]\(-1, 1, -1, 1, \ldots\)[/tex] is Neither.
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7,\ldots\)[/tex]
- Arithmetic Check: In an arithmetic sequence, the difference between consecutive terms is constant. Here, the differences are:
- [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 the same, this sequence is arithmetic.
- Geometric Check: In a geometric sequence, the ratio of consecutive terms must be constant, which is not true in this case.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Arithmetic Check: The differences between terms vary, so it's not arithmetic.
- Geometric Check: Ratios also vary between terms, so it's not geometric.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Arithmetic Check: Differences between terms are not constant, so it's not arithmetic.
- Geometric Check: The ratio of each term to the previous one is constant:
- [tex]\(\frac{3.5}{1.75} = 2\)[/tex]
- [tex]\(\frac{7}{3.5} = 2\)[/tex]
- [tex]\(\frac{14}{7} = 2\)[/tex]
Since the ratios are the same, this sequence is geometric.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Arithmetic Check: 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 constant, this sequence is arithmetic.
- Geometric Check: Ratios are not constant, so it's not geometric.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Arithmetic Check: The differences between terms are not constant, so it's not arithmetic.
- Geometric Check: Ratios are not constant either; it keeps alternating, so it's not geometric.
Now, summarizing the sequences:
- Sequence [tex]\(98.3, 94.1, 89.9, 85.7,\ldots\)[/tex] is Arithmetic.
- Sequence [tex]\(1, 0, -1, 0, \ldots\)[/tex] is Neither.
- Sequence [tex]\(1.75, 3.5, 7, 14\)[/tex] is Geometric.
- Sequence [tex]\(-12, -10.8, -9.6, -8.4\)[/tex] is Arithmetic.
- Sequence [tex]\(-1, 1, -1, 1, \ldots\)[/tex] is Neither.
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