Answer :
Certainly! Let's take a look at each sequence and determine whether it is arithmetic, geometric, or neither.
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Arithmetic Check: In an arithmetic sequence, the difference between consecutive terms is constant. Let's check 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 difference is consistent, this sequence is arithmetic.
- Geometric Check: Not needed, since it's already confirmed as arithmetic.
Conclusion: Arithmetic
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Arithmetic Check: Check the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are not the same, so it's not arithmetic.
- Geometric Check: Check the ratios:
- [tex]\(0/1 = 0\)[/tex] (division by zero in the next steps)
- [tex]\(-1/0\)[/tex] is undefined.
Because of division by zero, it can't be geometric.
Conclusion: Neither
3. Sequence 3: [tex]\(1.75, 35, 7, 14\)[/tex]
- Arithmetic Check: Differences are:
- [tex]\(35 - 1.75\)[/tex] is not consistent with other differences, so not arithmetic.
- Geometric Check: Ratios are:
- [tex]\(35/1.75 = 20\)[/tex]
- [tex]\(7/35 = 0.2\)[/tex]
The ratios are not the same, thus not geometric.
Conclusion: Neither
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Arithmetic Check: Differences are:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
The differences are constant, so this sequence is arithmetic.
- Geometric Check: Not necessary as it's already arithmetic.
Conclusion: Arithmetic
5. Sequence 5: [tex]\(-1, 1, -1, 1\)[/tex]
- Arithmetic Check: Differences are:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
Differences do not follow a constant pattern.
- Geometric Check: Ratios are:
- [tex]\(1/(-1) = -1\)[/tex]
- [tex]\(-1/1 = -1\)[/tex]
- [tex]\(1/(-1) = -1\)[/tex]
The ratios are constant, indicating that this could be geometric.
Conclusion: Geometric
So, here is the categorization of the sequences:
- Arithmetic: Sequence 1 ([tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]), Sequence 4 ([tex]\(-12, -10.8, -9.6, -8.4\)[/tex])
- Geometric: Sequence 5 ([tex]\(-1, 1, -1, 1\)[/tex])
- Neither: Sequence 2 ([tex]\(1, 0, -1, 0, \ldots\)[/tex]), Sequence 3 ([tex]\(1.75, 35, 7, 14\)[/tex])
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Arithmetic Check: In an arithmetic sequence, the difference between consecutive terms is constant. Let's check 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 difference is consistent, this sequence is arithmetic.
- Geometric Check: Not needed, since it's already confirmed as arithmetic.
Conclusion: Arithmetic
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Arithmetic Check: Check the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are not the same, so it's not arithmetic.
- Geometric Check: Check the ratios:
- [tex]\(0/1 = 0\)[/tex] (division by zero in the next steps)
- [tex]\(-1/0\)[/tex] is undefined.
Because of division by zero, it can't be geometric.
Conclusion: Neither
3. Sequence 3: [tex]\(1.75, 35, 7, 14\)[/tex]
- Arithmetic Check: Differences are:
- [tex]\(35 - 1.75\)[/tex] is not consistent with other differences, so not arithmetic.
- Geometric Check: Ratios are:
- [tex]\(35/1.75 = 20\)[/tex]
- [tex]\(7/35 = 0.2\)[/tex]
The ratios are not the same, thus not geometric.
Conclusion: Neither
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Arithmetic Check: Differences are:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
The differences are constant, so this sequence is arithmetic.
- Geometric Check: Not necessary as it's already arithmetic.
Conclusion: Arithmetic
5. Sequence 5: [tex]\(-1, 1, -1, 1\)[/tex]
- Arithmetic Check: Differences are:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
Differences do not follow a constant pattern.
- Geometric Check: Ratios are:
- [tex]\(1/(-1) = -1\)[/tex]
- [tex]\(-1/1 = -1\)[/tex]
- [tex]\(1/(-1) = -1\)[/tex]
The ratios are constant, indicating that this could be geometric.
Conclusion: Geometric
So, here is the categorization of the sequences:
- Arithmetic: Sequence 1 ([tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]), Sequence 4 ([tex]\(-12, -10.8, -9.6, -8.4\)[/tex])
- Geometric: Sequence 5 ([tex]\(-1, 1, -1, 1\)[/tex])
- Neither: Sequence 2 ([tex]\(1, 0, -1, 0, \ldots\)[/tex]), Sequence 3 ([tex]\(1.75, 35, 7, 14\)[/tex])
