Answer :
Let's go through each sequence to determine whether it is arithmetic, geometric, or neither.
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
To determine if a sequence is arithmetic, we check if the difference between consecutive terms is constant. Let's look at the differences:
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
All differences are the same (-4.2), so this sequence is an arithmetic sequence.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
First, we check for an arithmetic sequence by examining the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are not consistent, so it is not arithmetic.
Next, we check for a geometric sequence by examining the ratios:
- Ratios cannot be consistently calculated due to zeros in between, making it impossible for it to be geometric.
Since it is neither arithmetic nor geometric, this sequence is neither.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
We check for a geometric sequence by calculating the ratios between consecutive terms:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
All ratios are the same (2), so this sequence is a geometric sequence.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
Check for an arithmetic sequence by looking at 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]
All differences are constant (1.2), so this sequence is an arithmetic sequence.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
Check for arithmetic:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
The differences are not consistent, so it is not arithmetic.
Check for geometric:
- Not possible due to alternating signs and zero results in division, making it non-geometric.
This sequence is neither arithmetic nor geometric.
Summary:
- [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] is arithmetic.
- [tex]\(1, 0, -1, 0, \ldots\)[/tex] is neither.
- [tex]\(1.75, 3.5, 7, 14\)[/tex] is geometric.
- [tex]\(-12, -10.8, -9.6, -8.4\)[/tex] is arithmetic.
- [tex]\(-1, 1, -1, 1, \ldots\)[/tex] is neither.
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
To determine if a sequence is arithmetic, we check if the difference between consecutive terms is constant. Let's look at the differences:
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
All differences are the same (-4.2), so this sequence is an arithmetic sequence.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
First, we check for an arithmetic sequence by examining the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are not consistent, so it is not arithmetic.
Next, we check for a geometric sequence by examining the ratios:
- Ratios cannot be consistently calculated due to zeros in between, making it impossible for it to be geometric.
Since it is neither arithmetic nor geometric, this sequence is neither.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
We check for a geometric sequence by calculating the ratios between consecutive terms:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
All ratios are the same (2), so this sequence is a geometric sequence.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
Check for an arithmetic sequence by looking at 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]
All differences are constant (1.2), so this sequence is an arithmetic sequence.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
Check for arithmetic:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
The differences are not consistent, so it is not arithmetic.
Check for geometric:
- Not possible due to alternating signs and zero results in division, making it non-geometric.
This sequence is neither arithmetic nor geometric.
Summary:
- [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] is arithmetic.
- [tex]\(1, 0, -1, 0, \ldots\)[/tex] is neither.
- [tex]\(1.75, 3.5, 7, 14\)[/tex] is geometric.
- [tex]\(-12, -10.8, -9.6, -8.4\)[/tex] is arithmetic.
- [tex]\(-1, 1, -1, 1, \ldots\)[/tex] is neither.
