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Answer :
Let's analyze each sequence step-by-step to determine whether they are arithmetic, geometric, or neither.
1. Sequence 1: [tex]$98.3, 94.1, 89.9, 85.7, \ldots$[/tex]
To check if a sequence is arithmetic, 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 common difference is [tex]\(-4.2\)[/tex] for all steps, this sequence is Arithmetic.
2. Sequence 2: [tex]$1, 0, -1, 0, \ldots$[/tex]
For an arithmetic sequence, the difference should be constant, and for a geometric sequence, the ratio should be constant. Let's check:
Differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
Ratios:
- [tex]\(0/1\)[/tex] is undefined because division by zero is not possible.
Since neither the difference nor the ratio is consistent, this sequence is Neither.
3. Sequence 3: [tex]$1.75, 3.5, 7, 14$[/tex]
To check if a sequence is geometric, we look for a common ratio:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
Since the ratio is consistently [tex]\(2\)[/tex], this sequence is Geometric.
4. Sequence 4: [tex]$-12, -10.8, -9.6, -8.4$[/tex]
Calculate the differences to see 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 common difference is [tex]\(1.2\)[/tex], making this sequence Arithmetic.
5. Sequence 5: [tex]$-1, 1, -1, 1, \ldots$[/tex]
Checking for a common difference or ratio:
Differences:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
Ratios cannot be computed due to alternating negative and positive signs leading to inconsistent results, so the pattern breaks for a geometric sequence. This sequence does not have a consistent pattern for either arithmetic or geometric classification, so it is Neither.
So, the classifications are:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Geometric
- Sequence 4: Arithmetic
- Sequence 5: Neither
1. Sequence 1: [tex]$98.3, 94.1, 89.9, 85.7, \ldots$[/tex]
To check if a sequence is arithmetic, 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 common difference is [tex]\(-4.2\)[/tex] for all steps, this sequence is Arithmetic.
2. Sequence 2: [tex]$1, 0, -1, 0, \ldots$[/tex]
For an arithmetic sequence, the difference should be constant, and for a geometric sequence, the ratio should be constant. Let's check:
Differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
Ratios:
- [tex]\(0/1\)[/tex] is undefined because division by zero is not possible.
Since neither the difference nor the ratio is consistent, this sequence is Neither.
3. Sequence 3: [tex]$1.75, 3.5, 7, 14$[/tex]
To check if a sequence is geometric, we look for a common ratio:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
Since the ratio is consistently [tex]\(2\)[/tex], this sequence is Geometric.
4. Sequence 4: [tex]$-12, -10.8, -9.6, -8.4$[/tex]
Calculate the differences to see 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 common difference is [tex]\(1.2\)[/tex], making this sequence Arithmetic.
5. Sequence 5: [tex]$-1, 1, -1, 1, \ldots$[/tex]
Checking for a common difference or ratio:
Differences:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
Ratios cannot be computed due to alternating negative and positive signs leading to inconsistent results, so the pattern breaks for a geometric sequence. This sequence does not have a consistent pattern for either arithmetic or geometric classification, so it is Neither.
So, the classifications are:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Geometric
- Sequence 4: Arithmetic
- Sequence 5: Neither
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