Answer :
Sure! Let's analyze each sequence one by one to determine if they are arithmetic, geometric, or neither.
### Sequence 1: 98.3, 94.1, 89.9, 85.7, ...
An arithmetic sequence has a common difference between consecutive terms. Let's check:
- Difference 1: [tex]\( 94.1 - 98.3 = -4.2 \)[/tex]
- Difference 2: [tex]\( 89.9 - 94.1 = -4.2 \)[/tex]
- Difference 3: [tex]\( 85.7 - 89.9 = -4.2 \)[/tex]
Since all the differences are the same, this is an arithmetic sequence.
### Sequence 2: 1, 0, -1, 0, ...
An arithmetic sequence requires a constant difference, but here, the differences alternate, meaning they aren't constant. A geometric sequence requires a constant ratio, which also isn't present here. This sequence alternates in a pattern without a consistent rule:
This is neither arithmetic nor geometric.
### Sequence 3: 1.75, 3.5, 7, 14, ...
A geometric sequence has a common ratio between consecutive terms. Let's calculate:
- Ratio 1: [tex]\( \frac{3.5}{1.75} = 2 \)[/tex]
- Ratio 2: [tex]\( \frac{7}{3.5} = 2 \)[/tex]
- Ratio 3: [tex]\( \frac{14}{7} = 2 \)[/tex]
Since all the ratios are the same, this is a geometric sequence.
### Sequence 4: -12, -10.8, -9.6, -8.4
Let's check for an arithmetic sequence, which involves having a consistent difference:
- Difference 1: [tex]\( -10.8 - (-12) = 1.2 \)[/tex]
- Difference 2: [tex]\( -9.6 - (-10.8) = 1.2 \)[/tex]
- Difference 3: [tex]\( -8.4 - (-9.6) = 1.2 \)[/tex]
All differences are the same, so this is an arithmetic sequence.
### Sequence 5: -1, 1, -1, 1, ...
This sequence alternates consistently between -1 and 1. It lacks a constant difference or ratio:
This is neither arithmetic nor geometric.
So, here's how the sequences are classified:
1. Arithmetic
2. Neither
3. Geometric
4. Arithmetic
5. Neither
### Sequence 1: 98.3, 94.1, 89.9, 85.7, ...
An arithmetic sequence has a common difference between consecutive terms. Let's check:
- Difference 1: [tex]\( 94.1 - 98.3 = -4.2 \)[/tex]
- Difference 2: [tex]\( 89.9 - 94.1 = -4.2 \)[/tex]
- Difference 3: [tex]\( 85.7 - 89.9 = -4.2 \)[/tex]
Since all the differences are the same, this is an arithmetic sequence.
### Sequence 2: 1, 0, -1, 0, ...
An arithmetic sequence requires a constant difference, but here, the differences alternate, meaning they aren't constant. A geometric sequence requires a constant ratio, which also isn't present here. This sequence alternates in a pattern without a consistent rule:
This is neither arithmetic nor geometric.
### Sequence 3: 1.75, 3.5, 7, 14, ...
A geometric sequence has a common ratio between consecutive terms. Let's calculate:
- Ratio 1: [tex]\( \frac{3.5}{1.75} = 2 \)[/tex]
- Ratio 2: [tex]\( \frac{7}{3.5} = 2 \)[/tex]
- Ratio 3: [tex]\( \frac{14}{7} = 2 \)[/tex]
Since all the ratios are the same, this is a geometric sequence.
### Sequence 4: -12, -10.8, -9.6, -8.4
Let's check for an arithmetic sequence, which involves having a consistent difference:
- Difference 1: [tex]\( -10.8 - (-12) = 1.2 \)[/tex]
- Difference 2: [tex]\( -9.6 - (-10.8) = 1.2 \)[/tex]
- Difference 3: [tex]\( -8.4 - (-9.6) = 1.2 \)[/tex]
All differences are the same, so this is an arithmetic sequence.
### Sequence 5: -1, 1, -1, 1, ...
This sequence alternates consistently between -1 and 1. It lacks a constant difference or ratio:
This is neither arithmetic nor geometric.
So, here's how the sequences are classified:
1. Arithmetic
2. Neither
3. Geometric
4. Arithmetic
5. Neither
