Answer :
Let's analyze each sequence individually to determine whether they are arithmetic, geometric, or neither:
1. Sequence 1: [tex]$98.3, 94.1, 89.9, 85.7, \ldots$[/tex]
- An arithmetic sequence has a common difference between consecutive terms. Checking 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]
Since the differences are the same, this sequence is arithmetic.
2. Sequence 2: [tex]$1, 0, -1, 0, \ldots$[/tex]
- An arithmetic sequence requires a constant difference, and a geometric sequence requires a constant ratio. Checking the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are not consistent, and neither is the ratio. Thus, this sequence is neither arithmetic nor geometric.
3. Sequence 3: [tex]$1.75, 3.5, 7, 14$[/tex]
- Checking for a geometric sequence by the ratio:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
The ratio is constant. So, this sequence is geometric.
4. Sequence 4: [tex]$-12, -10.8, -9.6, -8.4$[/tex]
- Checking for an arithmetic sequence:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
Since there's a constant difference, this sequence is arithmetic.
5. Sequence 5: [tex]$-1, 1, -1, 1, \ldots$[/tex]
- Checking the differences and ratios:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
The differences alternate and do not form a consistent pattern, and the sequence alternates signs. Thus, it is neither arithmetic nor geometric.
Based on our analysis, the sequences can be categorized as follows:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Geometric
- Sequence 4: Arithmetic
- Sequence 5: Neither
1. Sequence 1: [tex]$98.3, 94.1, 89.9, 85.7, \ldots$[/tex]
- An arithmetic sequence has a common difference between consecutive terms. Checking 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]
Since the differences are the same, this sequence is arithmetic.
2. Sequence 2: [tex]$1, 0, -1, 0, \ldots$[/tex]
- An arithmetic sequence requires a constant difference, and a geometric sequence requires a constant ratio. Checking the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are not consistent, and neither is the ratio. Thus, this sequence is neither arithmetic nor geometric.
3. Sequence 3: [tex]$1.75, 3.5, 7, 14$[/tex]
- Checking for a geometric sequence by the ratio:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
The ratio is constant. So, this sequence is geometric.
4. Sequence 4: [tex]$-12, -10.8, -9.6, -8.4$[/tex]
- Checking for an arithmetic sequence:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
Since there's a constant difference, this sequence is arithmetic.
5. Sequence 5: [tex]$-1, 1, -1, 1, \ldots$[/tex]
- Checking the differences and ratios:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
The differences alternate and do not form a consistent pattern, and the sequence alternates signs. Thus, it is neither arithmetic nor geometric.
Based on our analysis, the sequences can be categorized as follows:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Geometric
- Sequence 4: Arithmetic
- Sequence 5: Neither
