Answer :
To determine whether the given sequences are arithmetic, geometric, or neither, let's analyze each sequence step-by-step.
### Sequence 1: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
1. Arithmetic Sequence Check:
- Calculate the differences 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]
- The differences are all consistent ([tex]\(-4.2\)[/tex]). Therefore, this sequence is arithmetic.
2. Geometric Sequence Check:
- Calculate the ratios between consecutive terms:
- [tex]\( \frac{94.1}{98.3} \approx 0.957 \)[/tex]
- [tex]\( \frac{89.9}{94.1} \approx 0.955 \)[/tex]
- [tex]\( \frac{85.7}{89.9} \approx 0.953 \)[/tex]
- The ratios are not consistent. Therefore, this sequence is not geometric.
Since the sequence is arithmetic, we categorize it as:
- Arithmetic: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
### Sequence 2: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
1. Arithmetic Sequence Check:
- Calculate the differences 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]
- The differences are all consistent ([tex]\(1.2\)[/tex]). Therefore, this sequence is arithmetic.
2. Geometric Sequence Check:
- Calculate the ratios between consecutive terms:
- [tex]\( \frac{-10.8}{-12} = 0.9 \)[/tex]
- [tex]\( \frac{-9.6}{-10.8} \approx 0.889 \)[/tex]
- [tex]\( \frac{-8.4}{-9.6} \approx 0.875 \)[/tex]
- The ratios are not consistent. Therefore, this sequence is not geometric.
Since the sequence is arithmetic, we categorize it as:
- Arithmetic: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
### Sequence 3: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
1. Arithmetic Sequence Check:
- Calculate the differences between consecutive terms:
- [tex]\( 1 - (-1) = 2 \)[/tex]
- [tex]\( -1 - 1 = -2 \)[/tex]
- [tex]\( 1 - (-1) = 2 \)[/tex]
- The differences are not consistent. Therefore, this sequence is not arithmetic.
2. Geometric Sequence Check:
- Calculate the ratios between consecutive terms:
- [tex]\( \frac{1}{-1} = -1 \)[/tex]
- [tex]\( \frac{-1}{1} = -1 \)[/tex]
- [tex]\( \frac{1}{-1} = -1 \)[/tex]
- The ratios are all consistent ([tex]\(-1\)[/tex]). Therefore, this sequence is geometric.
Since the sequence is geometric, we categorize it as:
- Geometric: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
### Final Categorization:
- Arithmetic: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex] and [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Geometric: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- Neither: None of the sequences fit this category based on our analysis.
### Sequence 1: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
1. Arithmetic Sequence Check:
- Calculate the differences 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]
- The differences are all consistent ([tex]\(-4.2\)[/tex]). Therefore, this sequence is arithmetic.
2. Geometric Sequence Check:
- Calculate the ratios between consecutive terms:
- [tex]\( \frac{94.1}{98.3} \approx 0.957 \)[/tex]
- [tex]\( \frac{89.9}{94.1} \approx 0.955 \)[/tex]
- [tex]\( \frac{85.7}{89.9} \approx 0.953 \)[/tex]
- The ratios are not consistent. Therefore, this sequence is not geometric.
Since the sequence is arithmetic, we categorize it as:
- Arithmetic: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
### Sequence 2: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
1. Arithmetic Sequence Check:
- Calculate the differences 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]
- The differences are all consistent ([tex]\(1.2\)[/tex]). Therefore, this sequence is arithmetic.
2. Geometric Sequence Check:
- Calculate the ratios between consecutive terms:
- [tex]\( \frac{-10.8}{-12} = 0.9 \)[/tex]
- [tex]\( \frac{-9.6}{-10.8} \approx 0.889 \)[/tex]
- [tex]\( \frac{-8.4}{-9.6} \approx 0.875 \)[/tex]
- The ratios are not consistent. Therefore, this sequence is not geometric.
Since the sequence is arithmetic, we categorize it as:
- Arithmetic: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
### Sequence 3: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
1. Arithmetic Sequence Check:
- Calculate the differences between consecutive terms:
- [tex]\( 1 - (-1) = 2 \)[/tex]
- [tex]\( -1 - 1 = -2 \)[/tex]
- [tex]\( 1 - (-1) = 2 \)[/tex]
- The differences are not consistent. Therefore, this sequence is not arithmetic.
2. Geometric Sequence Check:
- Calculate the ratios between consecutive terms:
- [tex]\( \frac{1}{-1} = -1 \)[/tex]
- [tex]\( \frac{-1}{1} = -1 \)[/tex]
- [tex]\( \frac{1}{-1} = -1 \)[/tex]
- The ratios are all consistent ([tex]\(-1\)[/tex]). Therefore, this sequence is geometric.
Since the sequence is geometric, we categorize it as:
- Geometric: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
### Final Categorization:
- Arithmetic: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex] and [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Geometric: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- Neither: None of the sequences fit this category based on our analysis.
