Answer :
Sure! Let's analyze each sequence to determine whether it is arithmetic, geometric, or neither.
### 1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
For an arithmetic sequence, the difference between consecutive terms, called the common difference (d), is fixed.
- First difference: [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- Second difference: [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- Third difference: [tex]\(85.7 - 89.9 = -4.2\)[/tex]
Since the differences are constant, this sequence is Arithmetic.
### 2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
For an arithmetic sequence, the difference between consecutive terms should be constant.
- First difference: [tex]\(0 - 1 = -1\)[/tex]
- Second difference: [tex]\(-1 - 0 = -1\)[/tex]
- Third difference: [tex]\(0 - (-1) = 1\)[/tex]
Since the differences are not constant, it is not arithmetic.
For a geometric sequence, the ratio between consecutive terms, called the common ratio (r), should be constant.
- First ratio: [tex]\(0 / 1 = 0\)[/tex]
- Second ratio: [tex]\(-1 / 0 = \text{undefined}\)[/tex]
The sequence does not have a constant ratio either, making it Neither arithmetic nor geometric.
### 3. Sequence: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
For an arithmetic sequence:
- First difference: [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- Second difference: [tex]\(7 - 3.5 = 3.5\)[/tex]
- Third difference: [tex]\(14 - 7 = 7\)[/tex]
Differences are not constant, so it is not arithmetic.
For a geometric sequence:
- First ratio: [tex]\(3.5 / 1.75 = 2\)[/tex]
- Second ratio: [tex]\(7 / 3.5 = 2\)[/tex]
- Third ratio: [tex]\(14 / 7 = 2\)[/tex]
Since the ratios are constant, this sequence is Geometric.
### 4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
For an arithmetic sequence:
- First difference: [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- Second difference: [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- Third difference: [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
Since the differences are constant, this sequence is Arithmetic.
### 5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
For an arithmetic sequence:
- First difference: [tex]\(1 - (-1) = 2\)[/tex]
- Second difference: [tex]\(-1 - 1 = -2\)[/tex]
- Third difference: [tex]\(1 - (-1) = 2\)[/tex]
Differences alternate and are not constant, so it is not arithmetic.
For a geometric sequence:
- First ratio: [tex]\(1 / -1 = -1\)[/tex]
- Second ratio: [tex]\(-1 / 1 = -1\)[/tex]
- Third ratio: [tex]\(1 / -1 = -1\)[/tex]
Since the ratios are constant, this sequence is Geometric.
### Summary
- [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]: Arithmetic
- [tex]\(1, 0, -1, 0, \ldots\)[/tex]: Neither
- [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]: Geometric
- [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]: Arithmetic
- [tex]\(-1, 1, -1, 1, \ldots\)[/tex]: Geometric
### 1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
For an arithmetic sequence, the difference between consecutive terms, called the common difference (d), is fixed.
- First difference: [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- Second difference: [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- Third difference: [tex]\(85.7 - 89.9 = -4.2\)[/tex]
Since the differences are constant, this sequence is Arithmetic.
### 2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
For an arithmetic sequence, the difference between consecutive terms should be constant.
- First difference: [tex]\(0 - 1 = -1\)[/tex]
- Second difference: [tex]\(-1 - 0 = -1\)[/tex]
- Third difference: [tex]\(0 - (-1) = 1\)[/tex]
Since the differences are not constant, it is not arithmetic.
For a geometric sequence, the ratio between consecutive terms, called the common ratio (r), should be constant.
- First ratio: [tex]\(0 / 1 = 0\)[/tex]
- Second ratio: [tex]\(-1 / 0 = \text{undefined}\)[/tex]
The sequence does not have a constant ratio either, making it Neither arithmetic nor geometric.
### 3. Sequence: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
For an arithmetic sequence:
- First difference: [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- Second difference: [tex]\(7 - 3.5 = 3.5\)[/tex]
- Third difference: [tex]\(14 - 7 = 7\)[/tex]
Differences are not constant, so it is not arithmetic.
For a geometric sequence:
- First ratio: [tex]\(3.5 / 1.75 = 2\)[/tex]
- Second ratio: [tex]\(7 / 3.5 = 2\)[/tex]
- Third ratio: [tex]\(14 / 7 = 2\)[/tex]
Since the ratios are constant, this sequence is Geometric.
### 4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
For an arithmetic sequence:
- First difference: [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- Second difference: [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- Third difference: [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
Since the differences are constant, this sequence is Arithmetic.
### 5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
For an arithmetic sequence:
- First difference: [tex]\(1 - (-1) = 2\)[/tex]
- Second difference: [tex]\(-1 - 1 = -2\)[/tex]
- Third difference: [tex]\(1 - (-1) = 2\)[/tex]
Differences alternate and are not constant, so it is not arithmetic.
For a geometric sequence:
- First ratio: [tex]\(1 / -1 = -1\)[/tex]
- Second ratio: [tex]\(-1 / 1 = -1\)[/tex]
- Third ratio: [tex]\(1 / -1 = -1\)[/tex]
Since the ratios are constant, this sequence is Geometric.
### Summary
- [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]: Arithmetic
- [tex]\(1, 0, -1, 0, \ldots\)[/tex]: Neither
- [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]: Geometric
- [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]: Arithmetic
- [tex]\(-1, 1, -1, 1, \ldots\)[/tex]: Geometric
