Answer :
Sure! Let's identify which of the sequences are arithmetic, which are geometric, and which are neither.
### Definitions:
- Arithmetic Sequence: This is a sequence where the difference between consecutive terms is constant. For example, in the sequence [tex]\(a, a + d, a + 2d, \ldots\)[/tex], the common difference is [tex]\(d\)[/tex].
- Geometric Sequence: This is a sequence where the ratio between consecutive terms is constant. For example, in the sequence [tex]\(a, ar, ar^2, \ldots\)[/tex], the common ratio is [tex]\(r\)[/tex].
### Now, let's analyze each sequence:
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Check if it's arithmetic:
- [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 difference is constant, so this is an arithmetic sequence.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Check both:
- Arithmetic: Differences are [tex]\(0 - 1 = -1\)[/tex], [tex]\(-1 - 0 = -1\)[/tex], [tex]\(0 + 1 = 1\)[/tex] (alternating, not constant).
- Geometric: Ratios are [tex]\(0/1 = 0\)[/tex] (undefined for division by zero) and [tex]\(-1/0\)[/tex] (undefined).
- This sequence does not have a consistent difference or ratio, so it is neither.
3. Sequence: [tex]\(1.75, 3.5, 7.14\)[/tex]
- Check if it's geometric:
- Ratios are [tex]\(3.5/1.75 = 2\)[/tex] and [tex]\(7.14/3.5 \approx 2.04\)[/tex]
- The ratio is not constant, so this is neither.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check if it's arithmetic:
- Differences are [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- The difference is constant, so this is an arithmetic sequence.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Check both:
- Arithmetic: Differences alternate between [tex]\(2\)[/tex] and [tex]\(-2\)[/tex]
- Geometric: Ratios are [tex]\(-1/1 = -1\)[/tex], [tex]\(1/-1 = -1\)[/tex]
- It repeats with alternating terms, but it forms a consistent ratio of [tex]\(-1\)[/tex]. Therefore, it is a geometric sequence.
### Conclusion:
- Arithmetic Sequences: [tex]\([98.3, 94.1, 89.9, 85.7]\)[/tex] and [tex]\([-12, -10.8, -9.6, -8.4]\)[/tex]
- Geometric Sequences: [tex]\([-1, 1, -1, 1]\)[/tex]
- Neither: [tex]\([1, 0, -1, 0]\)[/tex] and [tex]\([1.75, 3.5, 7.14]\)[/tex]
### Definitions:
- Arithmetic Sequence: This is a sequence where the difference between consecutive terms is constant. For example, in the sequence [tex]\(a, a + d, a + 2d, \ldots\)[/tex], the common difference is [tex]\(d\)[/tex].
- Geometric Sequence: This is a sequence where the ratio between consecutive terms is constant. For example, in the sequence [tex]\(a, ar, ar^2, \ldots\)[/tex], the common ratio is [tex]\(r\)[/tex].
### Now, let's analyze each sequence:
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Check if it's arithmetic:
- [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 difference is constant, so this is an arithmetic sequence.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Check both:
- Arithmetic: Differences are [tex]\(0 - 1 = -1\)[/tex], [tex]\(-1 - 0 = -1\)[/tex], [tex]\(0 + 1 = 1\)[/tex] (alternating, not constant).
- Geometric: Ratios are [tex]\(0/1 = 0\)[/tex] (undefined for division by zero) and [tex]\(-1/0\)[/tex] (undefined).
- This sequence does not have a consistent difference or ratio, so it is neither.
3. Sequence: [tex]\(1.75, 3.5, 7.14\)[/tex]
- Check if it's geometric:
- Ratios are [tex]\(3.5/1.75 = 2\)[/tex] and [tex]\(7.14/3.5 \approx 2.04\)[/tex]
- The ratio is not constant, so this is neither.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check if it's arithmetic:
- Differences are [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- The difference is constant, so this is an arithmetic sequence.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Check both:
- Arithmetic: Differences alternate between [tex]\(2\)[/tex] and [tex]\(-2\)[/tex]
- Geometric: Ratios are [tex]\(-1/1 = -1\)[/tex], [tex]\(1/-1 = -1\)[/tex]
- It repeats with alternating terms, but it forms a consistent ratio of [tex]\(-1\)[/tex]. Therefore, it is a geometric sequence.
### Conclusion:
- Arithmetic Sequences: [tex]\([98.3, 94.1, 89.9, 85.7]\)[/tex] and [tex]\([-12, -10.8, -9.6, -8.4]\)[/tex]
- Geometric Sequences: [tex]\([-1, 1, -1, 1]\)[/tex]
- Neither: [tex]\([1, 0, -1, 0]\)[/tex] and [tex]\([1.75, 3.5, 7.14]\)[/tex]
