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Answer :
We will examine each sequence by checking for a common difference (arithmetic) or a common ratio (geometric).
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Sequence 1: [tex]\(1,\, 0,\, -1,\, 0,\, \ldots\)[/tex]
1. Calculate the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are [tex]\(-1\)[/tex], [tex]\(-1\)[/tex], and [tex]\(1\)[/tex]. Since they are not all equal, the sequence is not arithmetic.
2. Check for a common ratio:
- [tex]\(0 \div 1 = 0\)[/tex]
- [tex]\(-1 \div 0\)[/tex] is undefined because division by zero is not allowed.
With an undefined ratio, the sequence is not geometric.
Thus, Sequence 1 is Neither.
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Sequence 2: [tex]\(1.75,\, 3.5,\, 7,\, 14\)[/tex]
1. Calculate the ratios:
- [tex]\(\frac{3.5}{1.75} = 2\)[/tex]
- [tex]\(\frac{7}{3.5} = 2\)[/tex]
- [tex]\(\frac{14}{7} = 2\)[/tex]
As the common ratio is [tex]\(2\)[/tex] for all pairs, the sequence is geometric.
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Sequence 3: [tex]\(-12,\, -10.8,\, -9.6,\, -8.4\)[/tex]
1. Calculate 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 each pair of consecutive terms has the same difference [tex]\(1.2\)[/tex], the sequence is arithmetic.
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Sequence 4: [tex]\(98.3,\, 94.1,\, 89.9,\, 85.7,\, \ldots\)[/tex]
1. 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]
Here, the common difference is [tex]\(-4.2\)[/tex]. Therefore, the sequence is arithmetic.
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Sequence 5: [tex]\(-1,\, 1,\, -1,\, 1,\, \ldots\)[/tex]
1. Calculate the ratios:
- [tex]\(\frac{1}{-1} = -1\)[/tex]
- [tex]\(\frac{-1}{1} = -1\)[/tex]
- [tex]\(\frac{1}{-1} = -1\)[/tex]
The common ratio is [tex]\(-1\)[/tex] for all terms. Thus, the sequence is geometric.
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Final Classification:
- Sequence 1: Neither
- Sequence 2: Geometric
- Sequence 3: Arithmetic
- Sequence 4: Arithmetic
- Sequence 5: Geometric
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Sequence 1: [tex]\(1,\, 0,\, -1,\, 0,\, \ldots\)[/tex]
1. Calculate the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are [tex]\(-1\)[/tex], [tex]\(-1\)[/tex], and [tex]\(1\)[/tex]. Since they are not all equal, the sequence is not arithmetic.
2. Check for a common ratio:
- [tex]\(0 \div 1 = 0\)[/tex]
- [tex]\(-1 \div 0\)[/tex] is undefined because division by zero is not allowed.
With an undefined ratio, the sequence is not geometric.
Thus, Sequence 1 is Neither.
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Sequence 2: [tex]\(1.75,\, 3.5,\, 7,\, 14\)[/tex]
1. Calculate the ratios:
- [tex]\(\frac{3.5}{1.75} = 2\)[/tex]
- [tex]\(\frac{7}{3.5} = 2\)[/tex]
- [tex]\(\frac{14}{7} = 2\)[/tex]
As the common ratio is [tex]\(2\)[/tex] for all pairs, the sequence is geometric.
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Sequence 3: [tex]\(-12,\, -10.8,\, -9.6,\, -8.4\)[/tex]
1. Calculate 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 each pair of consecutive terms has the same difference [tex]\(1.2\)[/tex], the sequence is arithmetic.
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Sequence 4: [tex]\(98.3,\, 94.1,\, 89.9,\, 85.7,\, \ldots\)[/tex]
1. 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]
Here, the common difference is [tex]\(-4.2\)[/tex]. Therefore, the sequence is arithmetic.
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Sequence 5: [tex]\(-1,\, 1,\, -1,\, 1,\, \ldots\)[/tex]
1. Calculate the ratios:
- [tex]\(\frac{1}{-1} = -1\)[/tex]
- [tex]\(\frac{-1}{1} = -1\)[/tex]
- [tex]\(\frac{1}{-1} = -1\)[/tex]
The common ratio is [tex]\(-1\)[/tex] for all terms. Thus, the sequence is geometric.
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Final Classification:
- Sequence 1: Neither
- Sequence 2: Geometric
- Sequence 3: Arithmetic
- Sequence 4: Arithmetic
- Sequence 5: Geometric
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