Answer :
Sure! Let's sort the sequences based on whether they are arithmetic, geometric, or neither.
### Definitions:
- Arithmetic Sequence: The difference between consecutive terms is constant. This difference is called the common difference.
- Geometric Sequence: The ratio between consecutive terms is constant. This ratio is called the common ratio.
- Neither: The sequence does not meet the criteria for being arithmetic or geometric.
### Analysis of Each Sequence:
1. Sequence A: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Check if it's arithmetic: Calculate the difference between each pair of 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 difference is constant, this sequence is arithmetic.
2. Sequence B: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Check if it's arithmetic: Differences between terms:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- The differences aren't constant. Check if it's geometric:
- Ratios:
- [tex]\(0/1 = 0\)[/tex] (not defined for geometric as denominator should not be zero)
- Neither the differences nor the ratios are consistent, so this sequence is neither arithmetic nor geometric.
3. Sequence C: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Check if it's arithmetic: Differences between terms:
- [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- [tex]\(7 - 3.5 = 3.5\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
- Differences aren't constant. Check if it's geometric:
- Ratios:
- [tex]\(3.5/1.75 = 2\)[/tex]
- [tex]\(7/3.5 = 2\)[/tex]
- [tex]\(14/7 = 2\)[/tex]
- The ratio is constant, so this sequence is geometric.
4. Sequence D: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check if it's arithmetic: Differences between terms:
- [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 difference is constant, this sequence is arithmetic.
5. Sequence E: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Check if it's arithmetic: Differences between terms:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- Differences aren't constant. Check if it's geometric:
- Ratios:
- [tex]\(1/(-1) = -1\)[/tex]
- [tex]\(-1/1 = -1\)[/tex]
- [tex]\(1/(-1) = -1\)[/tex]
- The ratio is constant, but an alternating sign doesn't generally define a common ratio for geometric sequences in basic contexts, so this sequence might still be considered as neither.
### Conclusion:
- Arithmetic: A, D
- Geometric: C
- Neither: B, E
If you have more questions or need further explanation, feel free to ask!
### Definitions:
- Arithmetic Sequence: The difference between consecutive terms is constant. This difference is called the common difference.
- Geometric Sequence: The ratio between consecutive terms is constant. This ratio is called the common ratio.
- Neither: The sequence does not meet the criteria for being arithmetic or geometric.
### Analysis of Each Sequence:
1. Sequence A: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Check if it's arithmetic: Calculate the difference between each pair of 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 difference is constant, this sequence is arithmetic.
2. Sequence B: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Check if it's arithmetic: Differences between terms:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- The differences aren't constant. Check if it's geometric:
- Ratios:
- [tex]\(0/1 = 0\)[/tex] (not defined for geometric as denominator should not be zero)
- Neither the differences nor the ratios are consistent, so this sequence is neither arithmetic nor geometric.
3. Sequence C: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Check if it's arithmetic: Differences between terms:
- [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- [tex]\(7 - 3.5 = 3.5\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
- Differences aren't constant. Check if it's geometric:
- Ratios:
- [tex]\(3.5/1.75 = 2\)[/tex]
- [tex]\(7/3.5 = 2\)[/tex]
- [tex]\(14/7 = 2\)[/tex]
- The ratio is constant, so this sequence is geometric.
4. Sequence D: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check if it's arithmetic: Differences between terms:
- [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 difference is constant, this sequence is arithmetic.
5. Sequence E: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Check if it's arithmetic: Differences between terms:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- Differences aren't constant. Check if it's geometric:
- Ratios:
- [tex]\(1/(-1) = -1\)[/tex]
- [tex]\(-1/1 = -1\)[/tex]
- [tex]\(1/(-1) = -1\)[/tex]
- The ratio is constant, but an alternating sign doesn't generally define a common ratio for geometric sequences in basic contexts, so this sequence might still be considered as neither.
### Conclusion:
- Arithmetic: A, D
- Geometric: C
- Neither: B, E
If you have more questions or need further explanation, feel free to ask!
