Answer :
Sure, let's sort each sequence to determine if it is arithmetic, geometric, or neither!
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7\)[/tex]
- Arithmetic Sequence: Check if the difference between consecutive terms is the same.
- Difference: [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- Difference: [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- Difference: [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- Since all differences are equal, this is an arithmetic sequence.
2. Sequence: [tex]\(1, 0, -1, 0\)[/tex]
- Arithmetic Sequence: Check differences. They should be constant if the sequence is arithmetic.
- Difference: [tex]\(0 - 1 = -1\)[/tex]
- Difference: [tex]\(-1 - 0 = -1\)[/tex]
- Difference: [tex]\(0 - (-1) = 1\)[/tex]
- Differences are not constant.
- Geometric Sequence: Check ratios. They should be constant if the sequence is geometric.
- Ratio: [tex]\(0/1 = 0\)[/tex] (not defined for division by zero)
- None of the ratios calculated are equal or defined properly.
- Thus, this sequence is neither arithmetic nor geometric.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Geometric Sequence: Check if the ratio between consecutive terms is the same.
- Ratio: [tex]\(3.5 / 1.75 = 2\)[/tex]
- Ratio: [tex]\(7 / 3.5 = 2\)[/tex]
- Ratio: [tex]\(14 / 7 = 2\)[/tex]
- Since all ratios are equal, this is a geometric sequence.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Arithmetic Sequence: Check if the difference between consecutive terms is the same.
- Difference: [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- Difference: [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- Difference: [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- Since all differences are equal, this is an arithmetic sequence.
5. Sequence: [tex]\(-1, 1, -1, 1\)[/tex]
- Arithmetic Sequence: Check differences.
- Difference: [tex]\(1 - (-1) = 2\)[/tex]
- Difference: [tex]\(-1 - 1 = -2\)[/tex]
- Difference: [tex]\(1 - (-1) = 2\)[/tex]
- Differences are not constant.
- Geometric Sequence: Check ratios.
- Ratio: [tex]\(1/(-1) = -1\)[/tex]
- Ratio: [tex]\(-1/1 = -1\)[/tex]
- Ratios involve negative and zero terms complicating constancy in sign.
- This sequence is neither arithmetic nor geometric.
To sum up:
- Sequence [tex]\(98.3, 94.1, 89.9, 85.7\)[/tex] is arithmetic.
- Sequence [tex]\(1, 0, -1, 0\)[/tex] is neither.
- Sequence [tex]\(1.75, 3.5, 7, 14\)[/tex] is geometric.
- Sequence [tex]\(-12, -10.8, -9.6, -8.4\)[/tex] is arithmetic.
- Sequence [tex]\(-1, 1, -1, 1\)[/tex] is neither.
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7\)[/tex]
- Arithmetic Sequence: Check if the difference between consecutive terms is the same.
- Difference: [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- Difference: [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- Difference: [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- Since all differences are equal, this is an arithmetic sequence.
2. Sequence: [tex]\(1, 0, -1, 0\)[/tex]
- Arithmetic Sequence: Check differences. They should be constant if the sequence is arithmetic.
- Difference: [tex]\(0 - 1 = -1\)[/tex]
- Difference: [tex]\(-1 - 0 = -1\)[/tex]
- Difference: [tex]\(0 - (-1) = 1\)[/tex]
- Differences are not constant.
- Geometric Sequence: Check ratios. They should be constant if the sequence is geometric.
- Ratio: [tex]\(0/1 = 0\)[/tex] (not defined for division by zero)
- None of the ratios calculated are equal or defined properly.
- Thus, this sequence is neither arithmetic nor geometric.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Geometric Sequence: Check if the ratio between consecutive terms is the same.
- Ratio: [tex]\(3.5 / 1.75 = 2\)[/tex]
- Ratio: [tex]\(7 / 3.5 = 2\)[/tex]
- Ratio: [tex]\(14 / 7 = 2\)[/tex]
- Since all ratios are equal, this is a geometric sequence.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Arithmetic Sequence: Check if the difference between consecutive terms is the same.
- Difference: [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- Difference: [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- Difference: [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- Since all differences are equal, this is an arithmetic sequence.
5. Sequence: [tex]\(-1, 1, -1, 1\)[/tex]
- Arithmetic Sequence: Check differences.
- Difference: [tex]\(1 - (-1) = 2\)[/tex]
- Difference: [tex]\(-1 - 1 = -2\)[/tex]
- Difference: [tex]\(1 - (-1) = 2\)[/tex]
- Differences are not constant.
- Geometric Sequence: Check ratios.
- Ratio: [tex]\(1/(-1) = -1\)[/tex]
- Ratio: [tex]\(-1/1 = -1\)[/tex]
- Ratios involve negative and zero terms complicating constancy in sign.
- This sequence is neither arithmetic nor geometric.
To sum up:
- Sequence [tex]\(98.3, 94.1, 89.9, 85.7\)[/tex] is arithmetic.
- Sequence [tex]\(1, 0, -1, 0\)[/tex] is neither.
- Sequence [tex]\(1.75, 3.5, 7, 14\)[/tex] is geometric.
- Sequence [tex]\(-12, -10.8, -9.6, -8.4\)[/tex] is arithmetic.
- Sequence [tex]\(-1, 1, -1, 1\)[/tex] is neither.
