Answer :
Sure, let's examine each sequence step by step to determine whether they are arithmetic, geometric, or neither.
### Sequence 1: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
1. Arithmetic sequence check: In an arithmetic sequence, the difference between each term and the previous one is constant. Here, [tex]\(0 - 1 = -1\)[/tex], [tex]\(-1 - 0 = -1\)[/tex], and [tex]\(0 - (-1) = 1\)[/tex]. Since the differences are not constant, it is not an arithmetic sequence.
2. Geometric sequence check: In a geometric sequence, the ratio between each term and the previous one is constant. Here, [tex]\(0/1 = 0\)[/tex], [tex]\(-1/0\)[/tex] is undefined, so it is not geometric.
This sequence is neither arithmetic nor geometric.
### Sequence 2: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
1. Arithmetic sequence check: The differences are [tex]\(1 - (-1) = 2\)[/tex], [tex]\(-1 - 1 = -2\)[/tex], and [tex]\(1 - (-1) = 2\)[/tex]. The differences are not constant.
2. Geometric sequence check: The ratios are [tex]\(1/(-1) = -1\)[/tex], [tex]\(-1/1 = -1\)[/tex], and [tex]\(1/(-1) = -1\)[/tex]. Although the ratio is constant, this pattern alternates and the ratio is negative. Typically, geometric sequences have a constant positive or negative ratio, not alternating signs.
This sequence is neither arithmetic nor geometric.
### Sequence 3: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
1. Arithmetic sequence check: The differences are [tex]\(-10.8 - (-12) = 1.2\)[/tex], [tex]\(-9.6 - (-10.8) = 1.2\)[/tex], and [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]. The differences are constant.
2. Geometric sequence check: The ratios are [tex]\(-10.8 / -12 \approx 0.9\)[/tex], [tex]\(-9.6 / -10.8 \approx 0.89\)[/tex], and [tex]\(-8.4 / -9.6 \approx 0.88\)[/tex]. The ratios are not constant.
This sequence is an arithmetic sequence.
### Sequence 4: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
1. Arithmetic sequence check: The differences are [tex]\(94.1 - 98.3 = -4.2\)[/tex], [tex]\(89.9 - 94.1 = -4.2\)[/tex], and [tex]\(85.7 - 89.9 = -4.2\)[/tex]. The differences are constant.
2. Geometric sequence check: The ratios are [tex]\(94.1 / 98.3 \approx 0.957\)[/tex], [tex]\(89.9 / 94.1 \approx 0.955\)[/tex], and [tex]\(85.7 / 89.9 \approx 0.953\)[/tex]. The ratios are not constant.
This sequence is an arithmetic sequence.
### Sequence 5: [tex]\(1.75, 3.5, 7, 14\)[/tex]
1. Arithmetic sequence check: The differences are [tex]\(3.5 - 1.75 = 1.75\)[/tex], [tex]\(7 - 3.5 = 3.5\)[/tex], and [tex]\(14 - 7 = 7\)[/tex]. The differences are not constant.
2. Geometric sequence check: The ratios are [tex]\(3.5 / 1.75 = 2\)[/tex], [tex]\(7 / 3.5 = 2\)[/tex], and [tex]\(14 / 7 = 2\)[/tex]. The ratios are constant.
This sequence is a geometric sequence.
### Summary:
- Sequence [tex]\(1,0,-1,0, \ldots\)[/tex] is neither.
- Sequence [tex]\(-1,1,-1,1, \ldots\)[/tex] is neither.
- Sequence [tex]\(-12,-10.8,-9.6,-8.4\)[/tex] is arithmetic.
- Sequence [tex]\(98.3,94.1,89.9,85.7, \ldots\)[/tex] is arithmetic.
- Sequence [tex]\(1.75,3.5,7,14\)[/tex] is geometric.
These evaluations categorize each sequence appropriately.
### Sequence 1: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
1. Arithmetic sequence check: In an arithmetic sequence, the difference between each term and the previous one is constant. Here, [tex]\(0 - 1 = -1\)[/tex], [tex]\(-1 - 0 = -1\)[/tex], and [tex]\(0 - (-1) = 1\)[/tex]. Since the differences are not constant, it is not an arithmetic sequence.
2. Geometric sequence check: In a geometric sequence, the ratio between each term and the previous one is constant. Here, [tex]\(0/1 = 0\)[/tex], [tex]\(-1/0\)[/tex] is undefined, so it is not geometric.
This sequence is neither arithmetic nor geometric.
### Sequence 2: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
1. Arithmetic sequence check: The differences are [tex]\(1 - (-1) = 2\)[/tex], [tex]\(-1 - 1 = -2\)[/tex], and [tex]\(1 - (-1) = 2\)[/tex]. The differences are not constant.
2. Geometric sequence check: The ratios are [tex]\(1/(-1) = -1\)[/tex], [tex]\(-1/1 = -1\)[/tex], and [tex]\(1/(-1) = -1\)[/tex]. Although the ratio is constant, this pattern alternates and the ratio is negative. Typically, geometric sequences have a constant positive or negative ratio, not alternating signs.
This sequence is neither arithmetic nor geometric.
### Sequence 3: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
1. Arithmetic sequence check: The differences are [tex]\(-10.8 - (-12) = 1.2\)[/tex], [tex]\(-9.6 - (-10.8) = 1.2\)[/tex], and [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]. The differences are constant.
2. Geometric sequence check: The ratios are [tex]\(-10.8 / -12 \approx 0.9\)[/tex], [tex]\(-9.6 / -10.8 \approx 0.89\)[/tex], and [tex]\(-8.4 / -9.6 \approx 0.88\)[/tex]. The ratios are not constant.
This sequence is an arithmetic sequence.
### Sequence 4: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
1. Arithmetic sequence check: The differences are [tex]\(94.1 - 98.3 = -4.2\)[/tex], [tex]\(89.9 - 94.1 = -4.2\)[/tex], and [tex]\(85.7 - 89.9 = -4.2\)[/tex]. The differences are constant.
2. Geometric sequence check: The ratios are [tex]\(94.1 / 98.3 \approx 0.957\)[/tex], [tex]\(89.9 / 94.1 \approx 0.955\)[/tex], and [tex]\(85.7 / 89.9 \approx 0.953\)[/tex]. The ratios are not constant.
This sequence is an arithmetic sequence.
### Sequence 5: [tex]\(1.75, 3.5, 7, 14\)[/tex]
1. Arithmetic sequence check: The differences are [tex]\(3.5 - 1.75 = 1.75\)[/tex], [tex]\(7 - 3.5 = 3.5\)[/tex], and [tex]\(14 - 7 = 7\)[/tex]. The differences are not constant.
2. Geometric sequence check: The ratios are [tex]\(3.5 / 1.75 = 2\)[/tex], [tex]\(7 / 3.5 = 2\)[/tex], and [tex]\(14 / 7 = 2\)[/tex]. The ratios are constant.
This sequence is a geometric sequence.
### Summary:
- Sequence [tex]\(1,0,-1,0, \ldots\)[/tex] is neither.
- Sequence [tex]\(-1,1,-1,1, \ldots\)[/tex] is neither.
- Sequence [tex]\(-12,-10.8,-9.6,-8.4\)[/tex] is arithmetic.
- Sequence [tex]\(98.3,94.1,89.9,85.7, \ldots\)[/tex] is arithmetic.
- Sequence [tex]\(1.75,3.5,7,14\)[/tex] is geometric.
These evaluations categorize each sequence appropriately.
