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Answer :
Let's determine whether each sequence is arithmetic, geometric, or neither by analyzing the patterns in the sequences.
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To check if this is an arithmetic sequence, we calculate the differences between consecutive terms:
- [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 with a common difference of [tex]\(-4.2\)[/tex].
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- To check if this is an arithmetic sequence:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are not the same.
- To check if this is a geometric sequence, we calculate the ratios:
- [tex]\(0/1 = 0\)[/tex] (division by zero is undefined from the next term)
The ratios are not consistent either.
Therefore, this sequence is neither arithmetic nor geometric.
3. Sequence 3: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- To check if this is a geometric sequence, we calculate the ratios:
- [tex]\(3.5/1.75 = 2\)[/tex]
- [tex]\(7/3.5 = 2\)[/tex]
- [tex]\(14/7 = 2\)[/tex]
Since the ratios are the same, this sequence is geometric with a common ratio of [tex]\(2\)[/tex].
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- For an arithmetic check, 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 the differences are the same, this sequence is arithmetic with a common difference of [tex]\(1.2\)[/tex].
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- For an arithmetic check:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
The differences aren't consistent.
- For a geometric check:
- [tex]\(1/(-1) = -1\)[/tex]
- [tex]\(-1/1 = -1\)[/tex]
- [tex]\(1/(-1) = -1\)[/tex]
The ratios are the same, so this sequence is geometric with a common ratio of [tex]\(-1\)[/tex].
Summing up each sequence:
- Sequence 1 is arithmetic.
- Sequence 2 is neither.
- Sequence 3 is geometric.
- Sequence 4 is arithmetic.
- Sequence 5 is geometric.
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To check if this is an arithmetic sequence, we calculate the differences between consecutive terms:
- [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 with a common difference of [tex]\(-4.2\)[/tex].
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- To check if this is an arithmetic sequence:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are not the same.
- To check if this is a geometric sequence, we calculate the ratios:
- [tex]\(0/1 = 0\)[/tex] (division by zero is undefined from the next term)
The ratios are not consistent either.
Therefore, this sequence is neither arithmetic nor geometric.
3. Sequence 3: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- To check if this is a geometric sequence, we calculate the ratios:
- [tex]\(3.5/1.75 = 2\)[/tex]
- [tex]\(7/3.5 = 2\)[/tex]
- [tex]\(14/7 = 2\)[/tex]
Since the ratios are the same, this sequence is geometric with a common ratio of [tex]\(2\)[/tex].
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- For an arithmetic check, 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 the differences are the same, this sequence is arithmetic with a common difference of [tex]\(1.2\)[/tex].
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- For an arithmetic check:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
The differences aren't consistent.
- For a geometric check:
- [tex]\(1/(-1) = -1\)[/tex]
- [tex]\(-1/1 = -1\)[/tex]
- [tex]\(1/(-1) = -1\)[/tex]
The ratios are the same, so this sequence is geometric with a common ratio of [tex]\(-1\)[/tex].
Summing up each sequence:
- Sequence 1 is arithmetic.
- Sequence 2 is neither.
- Sequence 3 is geometric.
- Sequence 4 is arithmetic.
- Sequence 5 is geometric.
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