Answer :
Let's analyze each sequence and determine whether it is arithmetic, geometric, or neither by checking the common differences or ratios.
1. Sequence: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
- Check for Arithmetic: An arithmetic sequence has a common difference between consecutive terms.
- Difference: [tex]\( 94.1 - 98.3 = -4.2 \)[/tex]
- Difference: [tex]\( 89.9 - 94.1 = -4.2 \)[/tex]
- Difference: [tex]\( 85.7 - 89.9 = -4.2 \)[/tex]
- Since the common difference is the same, this sequence is arithmetic.
2. Sequence: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
- Check for Arithmetic:
- Difference: [tex]\( 0 - 1 = -1 \)[/tex]
- Difference: [tex]\( -1 - 0 = -1 \)[/tex]
- Difference: [tex]\( 0 - (-1) = 1 \)[/tex]
- The differences are not consistent, so it is not arithmetic.
- Check for Geometric: A geometric sequence has a common ratio between consecutive terms. The ratios [tex]\( 0/1 \)[/tex], [tex]\(-1/0\)[/tex] (undefined), etc., are inconsistent. Thus, it is not geometric.
- This sequence is neither.
3. Sequence: [tex]\( 1.75, 3.5, 7, 14, \ldots \)[/tex]
- Check for Geometric:
- Ratio: [tex]\( 3.5/1.75 = 2 \)[/tex]
- Ratio: [tex]\( 7/3.5 = 2 \)[/tex]
- Ratio: [tex]\( 14/7 = 2 \)[/tex]
- Since the common ratio is the same, this sequence is geometric.
4. Sequence: [tex]\( -12, -10.8, -9.6, -8.4, \ldots \)[/tex]
- Check for Arithmetic:
- Difference: [tex]\( -10.8 - (-12) = 1.2 \)[/tex]
- Difference: [tex]\( -9.6 - (-10.8) = 1.2 \)[/tex]
- Difference: [tex]\( -8.4 - (-9.6) = 1.2 \)[/tex]
- Since the common difference is the same, this sequence is arithmetic.
5. Sequence: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- Check for Arithmetic:
- Difference: [tex]\( 1 - (-1) = 2 \)[/tex]
- Difference: [tex]\( -1 - 1 = -2 \)[/tex]
- Difference: [tex]\( 1 - (-1) = 2 \)[/tex]
- The differences are not consistent, so it is not arithmetic.
- Check for Geometric:
- Ratio: [tex]\( 1/(-1) = -1 \)[/tex]
- Ratio: [tex]\( -1/1 = -1 \)[/tex]
- Ratio: [tex]\( 1/(-1) = -1 \)[/tex]
- Since the common ratio is the same, this sequence is geometric.
To summarize:
- Arithmetic Sequences: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex] and [tex]\( -12, -10.8, -9.6, -8.4, \ldots \)[/tex]
- Geometric Sequences: [tex]\( 1.75, 3.5, 7, 14, \ldots \)[/tex] and [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- Neither: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
1. Sequence: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
- Check for Arithmetic: An arithmetic sequence has a common difference between consecutive terms.
- Difference: [tex]\( 94.1 - 98.3 = -4.2 \)[/tex]
- Difference: [tex]\( 89.9 - 94.1 = -4.2 \)[/tex]
- Difference: [tex]\( 85.7 - 89.9 = -4.2 \)[/tex]
- Since the common difference is the same, this sequence is arithmetic.
2. Sequence: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
- Check for Arithmetic:
- Difference: [tex]\( 0 - 1 = -1 \)[/tex]
- Difference: [tex]\( -1 - 0 = -1 \)[/tex]
- Difference: [tex]\( 0 - (-1) = 1 \)[/tex]
- The differences are not consistent, so it is not arithmetic.
- Check for Geometric: A geometric sequence has a common ratio between consecutive terms. The ratios [tex]\( 0/1 \)[/tex], [tex]\(-1/0\)[/tex] (undefined), etc., are inconsistent. Thus, it is not geometric.
- This sequence is neither.
3. Sequence: [tex]\( 1.75, 3.5, 7, 14, \ldots \)[/tex]
- Check for Geometric:
- Ratio: [tex]\( 3.5/1.75 = 2 \)[/tex]
- Ratio: [tex]\( 7/3.5 = 2 \)[/tex]
- Ratio: [tex]\( 14/7 = 2 \)[/tex]
- Since the common ratio is the same, this sequence is geometric.
4. Sequence: [tex]\( -12, -10.8, -9.6, -8.4, \ldots \)[/tex]
- Check for Arithmetic:
- Difference: [tex]\( -10.8 - (-12) = 1.2 \)[/tex]
- Difference: [tex]\( -9.6 - (-10.8) = 1.2 \)[/tex]
- Difference: [tex]\( -8.4 - (-9.6) = 1.2 \)[/tex]
- Since the common difference is the same, this sequence is arithmetic.
5. Sequence: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- Check for Arithmetic:
- Difference: [tex]\( 1 - (-1) = 2 \)[/tex]
- Difference: [tex]\( -1 - 1 = -2 \)[/tex]
- Difference: [tex]\( 1 - (-1) = 2 \)[/tex]
- The differences are not consistent, so it is not arithmetic.
- Check for Geometric:
- Ratio: [tex]\( 1/(-1) = -1 \)[/tex]
- Ratio: [tex]\( -1/1 = -1 \)[/tex]
- Ratio: [tex]\( 1/(-1) = -1 \)[/tex]
- Since the common ratio is the same, this sequence is geometric.
To summarize:
- Arithmetic Sequences: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex] and [tex]\( -12, -10.8, -9.6, -8.4, \ldots \)[/tex]
- Geometric Sequences: [tex]\( 1.75, 3.5, 7, 14, \ldots \)[/tex] and [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- Neither: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
