Answer :
Certainly! Let's analyze each sequence step-by-step to determine whether they are arithmetic, geometric, or neither.
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Arithmetic: Check if the difference between consecutive terms is constant.
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- The difference is constant ([tex]\(-4.2\)[/tex]), so this sequence is arithmetic.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Arithmetic: Check the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- The differences are not constant, so it's not arithmetic.
- Geometric: Check the ratios:
- [tex]\(0 / 1 = 0\)[/tex]
- [tex]\(-1 / 0\)[/tex] is undefined.
- Since [tex]\(0\)[/tex] in a ratio makes it undefined, this sequence is neither arithmetic nor geometric.
- Therefore, this sequence is neither.
3. Sequence: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- Arithmetic: Check the differences:
- [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- [tex]\(7 - 3.5 = 3.5\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
- The differences are not constant, so it is not arithmetic.
- Geometric: Check the ratios:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- The ratio is constant ([tex]\(2\)[/tex]), so this sequence is geometric.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Arithmetic: Check 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]
- The difference is constant ([tex]\(1.2\)[/tex]), so this sequence is arithmetic.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Arithmetic: Check the differences:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- The differences are not constant, so it’s not arithmetic.
- Geometric: Check the ratios:
- [tex]\(1 / (-1) = -1\)[/tex]
- [tex]\(-1 / 1 = -1\)[/tex]
- [tex]\(1 / (-1) = -1\)[/tex]
- The ratio is constant ([tex]\(-1\)[/tex]), so this sequence is geometric.
Now we can sort the sequences:
- Arithmetic Sequences:
- [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Geometric Sequences:
- [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Neither:
- [tex]\(1, 0, -1, 0, \ldots\)[/tex]
In conclusion:
- Arithmetic Sequences: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex].
- Geometric Sequences: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex] and [tex]\(-1, 1, -1, 1, \ldots\)[/tex].
- Neither: [tex]\(1, 0, -1, 0, \ldots\)[/tex].
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Arithmetic: Check if the difference between consecutive terms is constant.
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- The difference is constant ([tex]\(-4.2\)[/tex]), so this sequence is arithmetic.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Arithmetic: Check the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- The differences are not constant, so it's not arithmetic.
- Geometric: Check the ratios:
- [tex]\(0 / 1 = 0\)[/tex]
- [tex]\(-1 / 0\)[/tex] is undefined.
- Since [tex]\(0\)[/tex] in a ratio makes it undefined, this sequence is neither arithmetic nor geometric.
- Therefore, this sequence is neither.
3. Sequence: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- Arithmetic: Check the differences:
- [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- [tex]\(7 - 3.5 = 3.5\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
- The differences are not constant, so it is not arithmetic.
- Geometric: Check the ratios:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- The ratio is constant ([tex]\(2\)[/tex]), so this sequence is geometric.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Arithmetic: Check 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]
- The difference is constant ([tex]\(1.2\)[/tex]), so this sequence is arithmetic.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Arithmetic: Check the differences:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- The differences are not constant, so it’s not arithmetic.
- Geometric: Check the ratios:
- [tex]\(1 / (-1) = -1\)[/tex]
- [tex]\(-1 / 1 = -1\)[/tex]
- [tex]\(1 / (-1) = -1\)[/tex]
- The ratio is constant ([tex]\(-1\)[/tex]), so this sequence is geometric.
Now we can sort the sequences:
- Arithmetic Sequences:
- [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Geometric Sequences:
- [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Neither:
- [tex]\(1, 0, -1, 0, \ldots\)[/tex]
In conclusion:
- Arithmetic Sequences: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex].
- Geometric Sequences: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex] and [tex]\(-1, 1, -1, 1, \ldots\)[/tex].
- Neither: [tex]\(1, 0, -1, 0, \ldots\)[/tex].
