Answer :
Certainly! Let's analyze each sequence to determine whether it is arithmetic, geometric, or neither:
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Arithmetic Check: We find 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]
Since the differences are the same, this sequence is arithmetic.
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Arithmetic Check: Differences between consecutive terms:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- Geometric Check: Ratios are not consistent and the sequence repeats values, suggesting it is neither arithmetic nor geometric. This is an example of neither.
3. Sequence 3: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- Geometric Check: Ratios between consecutive terms:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
Since the ratios are consistent, this sequence is geometric.
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Arithmetic Check: 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]
Since the differences are the same, this sequence is arithmetic.
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Arithmetic Check: Differences alternate:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- Geometric Check: Ratios alternate:
- [tex]\(1 / (-1) = -1\)[/tex]
- [tex]\(-1 / 1 = -1\)[/tex]
- [tex]\(1 / (-1) = -1\)[/tex]
This sequence alternates signs but maintains a consistent ratio, hence it can be considered geometric with a common ratio of [tex]\(-1\)[/tex].
In summary, the sequences are classified as follows:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Geometric
- Sequence 4: Arithmetic
- Sequence 5: Geometric
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Arithmetic Check: We find 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]
Since the differences are the same, this sequence is arithmetic.
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Arithmetic Check: Differences between consecutive terms:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- Geometric Check: Ratios are not consistent and the sequence repeats values, suggesting it is neither arithmetic nor geometric. This is an example of neither.
3. Sequence 3: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- Geometric Check: Ratios between consecutive terms:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
Since the ratios are consistent, this sequence is geometric.
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Arithmetic Check: 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]
Since the differences are the same, this sequence is arithmetic.
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Arithmetic Check: Differences alternate:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- Geometric Check: Ratios alternate:
- [tex]\(1 / (-1) = -1\)[/tex]
- [tex]\(-1 / 1 = -1\)[/tex]
- [tex]\(1 / (-1) = -1\)[/tex]
This sequence alternates signs but maintains a consistent ratio, hence it can be considered geometric with a common ratio of [tex]\(-1\)[/tex].
In summary, the sequences are classified as follows:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Geometric
- Sequence 4: Arithmetic
- Sequence 5: Geometric
