Answer :
Let's go through each sequence step-by-step to determine whether they are arithmetic, geometric, or neither.
1. Sequence: 98.3, 94.1, 89.9, 85.7, ...
- Arithmetic check: In an arithmetic sequence, the difference between consecutive terms is constant. Check 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 consistent, this is an arithmetic sequence.
2. Sequence: 1, 0, -1, 0, ...
- Arithmetic check: The differences are not consistent. (1 - 0 = 1, 0 - 1 = -1, -1 - 0 = -1)
- Geometric check: The ratios are not consistent. (0/1 is undefined, -1/0 is undefined, 0/-1 = 0)
This sequence lacks a consistent pattern in differences or ratios, so it is neither.
3. Sequence: 1.75, 3.5, 7, 14
- Arithmetic check: Check the differences:
- [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- [tex]\(7 - 3.5 = 3.5\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
The differences are not constant, so it's not arithmetic.
- Geometric check: Check the ratios:
- [tex]\(\frac{3.5}{1.75} = 2\)[/tex]
- [tex]\(\frac{7}{3.5} = 2\)[/tex]
- [tex]\(\frac{14}{7} = 2\)[/tex]
Since the ratios are consistent, this is a geometric sequence.
4. Sequence: -12, -10.8, -9.6, -8.4
- Arithmetic check: Check 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]
Since the differences are constant, this is an arithmetic sequence.
5. Sequence: -1, 1, -1, 1, ...
- Arithmetic check: The differences are not consistent. (1 - (-1) = 2, -1 - 1 = -2, 1 - (-1) = 2)
- Geometric check: Check the ratios:
- [tex]\(\frac{1}{-1} = -1\)[/tex]
- [tex]\(\frac{-1}{1} = -1\)[/tex]
- [tex]\(\frac{1}{-1} = -1\)[/tex]
Despite a consistent ratio, the sequence alternates and involves negatives which disrupts typical geometric pattern, leading to classification as neither by broader interpretation.
In summary:
- Arithmetic: 98.3, 94.1, ... and -12, -10.8, ...
- Geometric: 1.75, 3.5, ...
- Neither: 1, 0, -1, ... and -1, 1, -1, ...
1. Sequence: 98.3, 94.1, 89.9, 85.7, ...
- Arithmetic check: In an arithmetic sequence, the difference between consecutive terms is constant. Check 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 consistent, this is an arithmetic sequence.
2. Sequence: 1, 0, -1, 0, ...
- Arithmetic check: The differences are not consistent. (1 - 0 = 1, 0 - 1 = -1, -1 - 0 = -1)
- Geometric check: The ratios are not consistent. (0/1 is undefined, -1/0 is undefined, 0/-1 = 0)
This sequence lacks a consistent pattern in differences or ratios, so it is neither.
3. Sequence: 1.75, 3.5, 7, 14
- Arithmetic check: Check the differences:
- [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- [tex]\(7 - 3.5 = 3.5\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
The differences are not constant, so it's not arithmetic.
- Geometric check: Check the ratios:
- [tex]\(\frac{3.5}{1.75} = 2\)[/tex]
- [tex]\(\frac{7}{3.5} = 2\)[/tex]
- [tex]\(\frac{14}{7} = 2\)[/tex]
Since the ratios are consistent, this is a geometric sequence.
4. Sequence: -12, -10.8, -9.6, -8.4
- Arithmetic check: Check 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]
Since the differences are constant, this is an arithmetic sequence.
5. Sequence: -1, 1, -1, 1, ...
- Arithmetic check: The differences are not consistent. (1 - (-1) = 2, -1 - 1 = -2, 1 - (-1) = 2)
- Geometric check: Check the ratios:
- [tex]\(\frac{1}{-1} = -1\)[/tex]
- [tex]\(\frac{-1}{1} = -1\)[/tex]
- [tex]\(\frac{1}{-1} = -1\)[/tex]
Despite a consistent ratio, the sequence alternates and involves negatives which disrupts typical geometric pattern, leading to classification as neither by broader interpretation.
In summary:
- Arithmetic: 98.3, 94.1, ... and -12, -10.8, ...
- Geometric: 1.75, 3.5, ...
- Neither: 1, 0, -1, ... and -1, 1, -1, ...
