Answer :
Of course! Let's identify each sequence one by one to determine whether they are arithmetic, geometric, or neither.
### Definitions:
1. Arithmetic Sequence: A sequence in which the difference between consecutive terms is constant. For example, in the sequence [tex]\(2, 5, 8, 11\)[/tex], the difference is [tex]\(3\)[/tex].
2. Geometric Sequence: A sequence in which the ratio of consecutive terms is constant. For example, in the sequence [tex]\(3, 6, 12, 24\)[/tex], the ratio is [tex]\(2\)[/tex].
### Analyze Each Sequence
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
Let's find the difference between 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: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
Let's find the difference between consecutive terms:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
Since the differences are not constant, let's check if it's geometric:
- [tex]\(0 / 1 = 0\)[/tex]
- [tex]\(-1 / 0 = \text{undefined}\)[/tex]
- [tex]\(0 / -1 = 0\)[/tex]
Since the ratios are not constant and sometimes undefined, this sequence is neither arithmetic nor geometric.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
Let's find the ratio between consecutive terms:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
Since the ratio is constant, this sequence is geometric.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
Let's find the difference between consecutive 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.
### Summary:
- [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] — Arithmetic
- [tex]\(1, 0, -1, 0, \ldots\)[/tex] — Neither
- [tex]\(1.75, 3.5, 7, 14\)[/tex] — Geometric
- [tex]\(-12, -10.8, -9.6, -8.4\)[/tex] — Arithmetic
I hope this breakdown helps! If you have any more questions, feel free to ask.
### Definitions:
1. Arithmetic Sequence: A sequence in which the difference between consecutive terms is constant. For example, in the sequence [tex]\(2, 5, 8, 11\)[/tex], the difference is [tex]\(3\)[/tex].
2. Geometric Sequence: A sequence in which the ratio of consecutive terms is constant. For example, in the sequence [tex]\(3, 6, 12, 24\)[/tex], the ratio is [tex]\(2\)[/tex].
### Analyze Each Sequence
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
Let's find the difference between 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: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
Let's find the difference between consecutive terms:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
Since the differences are not constant, let's check if it's geometric:
- [tex]\(0 / 1 = 0\)[/tex]
- [tex]\(-1 / 0 = \text{undefined}\)[/tex]
- [tex]\(0 / -1 = 0\)[/tex]
Since the ratios are not constant and sometimes undefined, this sequence is neither arithmetic nor geometric.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
Let's find the ratio between consecutive terms:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
Since the ratio is constant, this sequence is geometric.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
Let's find the difference between consecutive 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.
### Summary:
- [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] — Arithmetic
- [tex]\(1, 0, -1, 0, \ldots\)[/tex] — Neither
- [tex]\(1.75, 3.5, 7, 14\)[/tex] — Geometric
- [tex]\(-12, -10.8, -9.6, -8.4\)[/tex] — Arithmetic
I hope this breakdown helps! If you have any more questions, feel free to ask.
