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Answer :
Sure, let’s look at each sequence and determine whether it's arithmetic, geometric, or neither.
### Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
An arithmetic sequence has a constant difference between consecutive terms:
- Difference between [tex]\(94.1\)[/tex] and [tex]\(98.3\)[/tex] is [tex]\(94.1 - 98.3 = -4.2\)[/tex].
- Difference between [tex]\(89.9\)[/tex] and [tex]\(94.1\)[/tex] is [tex]\(89.9 - 94.1 = -4.2\)[/tex].
- Difference between [tex]\(85.7\)[/tex] and [tex]\(89.9\)[/tex] is [tex]\(85.7 - 89.9 = -4.2\)[/tex].
Since the difference is constant, this sequence is arithmetic.
### Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
Check if it's arithmetic:
- Difference between [tex]\(0\)[/tex] and [tex]\(1\)[/tex] is [tex]\(0 - 1 = -1\)[/tex].
- Difference between [tex]\(-1\)[/tex] and [tex]\(0\)[/tex] is [tex]\(-1 - 0 = -1\)[/tex].
- Difference between [tex]\(0\)[/tex] and [tex]\(-1\)[/tex] is [tex]\(0 - (-1) = 1\)[/tex].
Check if it's geometric:
- Ratio of [tex]\(0\)[/tex] to [tex]\(1\)[/tex] is [tex]\(0/1 = 0\)[/tex], which is not valid for a geometric sequence.
- Additionally, the sequence does not show a constant ratio.
Since it’s neither an arithmetic sequence nor a geometric sequence, it is neither.
### Sequence 3: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
Check if it's arithmetic:
- Difference between [tex]\(3.5\)[/tex] and [tex]\(1.75\)[/tex] is [tex]\(3.5 - 1.75 = 1.75\)[/tex].
- Difference between [tex]\(7\)[/tex] and [tex]\(3.5\)[/tex] is [tex]\(7 - 3.5 = 3.5\)[/tex].
- Difference between [tex]\(14\)[/tex] and [tex]\(7\)[/tex] is [tex]\(14 - 7 = 7\)[/tex].
Differences are not constant, so not arithmetic.
Check if it's geometric:
- Ratio of [tex]\(3.5\)[/tex] to [tex]\(1.75\)[/tex] is [tex]\(3.5 / 1.75 = 2\)[/tex].
- Ratio of [tex]\(7\)[/tex] to [tex]\(3.5\)[/tex] is [tex]\(7 / 3.5 = 2\)[/tex].
- Ratio of [tex]\(14\)[/tex] to [tex]\(7\)[/tex] is [tex]\(14 / 7 = 2\)[/tex].
Since the ratio is constant, this sequence is geometric.
### Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
An arithmetic sequence has a constant difference between consecutive terms:
- Difference between [tex]\(-10.8\)[/tex] and [tex]\(-12\)[/tex] is [tex]\(-10.8 - (-12) = 1.2\)[/tex].
- Difference between [tex]\(-9.6\)[/tex] and [tex]\(-10.8\)[/tex] is [tex]\(-9.6 - (-10.8) = 1.2\)[/tex].
- Difference between [tex]\(-8.4\)[/tex] and [tex]\(-9.6\)[/tex] is [tex]\(-8.4 - (-9.6) = 1.2\)[/tex].
Since the difference is constant, this sequence is arithmetic.
### Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
Check if it's arithmetic:
- Difference between [tex]\(1\)[/tex] and [tex]\(-1\)[/tex] is [tex]\(1 - (-1) = 2\)[/tex].
- Difference between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] is [tex]\(-1 - 1 = -2\)[/tex].
- Difference between [tex]\(1\)[/tex] and [tex]\(-1\)[/tex] is [tex]\(1 - (-1) = 2\)[/tex].
Check if it's geometric:
- Ratio of [tex]\(1\)[/tex] to [tex]\(-1\)[/tex] is [tex]\(1 / -1 = -1\)[/tex].
- Ratio of [tex]\(-1\)[/tex] to [tex]\(1\)[/tex] is [tex]\(-1 / 1 = -1\)[/tex].
- Ratio of [tex]\(1\)[/tex] to [tex]\(-1\)[/tex] is [tex]\(1 / -1 = -1\)[/tex].
Even though the ratios are constant, alternating signs generally suggest it's neither consistently growing nor shrinking.
Given the repeated pattern and switching signs, this sequence is neither arithmetic nor geometric.
### Summary
- [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]: Arithmetic
- [tex]\(1, 0, -1, 0, \ldots\)[/tex]: Neither
- [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]: Geometric
- [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]: Arithmetic
- [tex]\(-1, 1, -1, 1, \ldots\)[/tex]: Neither
I hope this helps! If you have any more questions, feel free to ask!
### Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
An arithmetic sequence has a constant difference between consecutive terms:
- Difference between [tex]\(94.1\)[/tex] and [tex]\(98.3\)[/tex] is [tex]\(94.1 - 98.3 = -4.2\)[/tex].
- Difference between [tex]\(89.9\)[/tex] and [tex]\(94.1\)[/tex] is [tex]\(89.9 - 94.1 = -4.2\)[/tex].
- Difference between [tex]\(85.7\)[/tex] and [tex]\(89.9\)[/tex] is [tex]\(85.7 - 89.9 = -4.2\)[/tex].
Since the difference is constant, this sequence is arithmetic.
### Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
Check if it's arithmetic:
- Difference between [tex]\(0\)[/tex] and [tex]\(1\)[/tex] is [tex]\(0 - 1 = -1\)[/tex].
- Difference between [tex]\(-1\)[/tex] and [tex]\(0\)[/tex] is [tex]\(-1 - 0 = -1\)[/tex].
- Difference between [tex]\(0\)[/tex] and [tex]\(-1\)[/tex] is [tex]\(0 - (-1) = 1\)[/tex].
Check if it's geometric:
- Ratio of [tex]\(0\)[/tex] to [tex]\(1\)[/tex] is [tex]\(0/1 = 0\)[/tex], which is not valid for a geometric sequence.
- Additionally, the sequence does not show a constant ratio.
Since it’s neither an arithmetic sequence nor a geometric sequence, it is neither.
### Sequence 3: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
Check if it's arithmetic:
- Difference between [tex]\(3.5\)[/tex] and [tex]\(1.75\)[/tex] is [tex]\(3.5 - 1.75 = 1.75\)[/tex].
- Difference between [tex]\(7\)[/tex] and [tex]\(3.5\)[/tex] is [tex]\(7 - 3.5 = 3.5\)[/tex].
- Difference between [tex]\(14\)[/tex] and [tex]\(7\)[/tex] is [tex]\(14 - 7 = 7\)[/tex].
Differences are not constant, so not arithmetic.
Check if it's geometric:
- Ratio of [tex]\(3.5\)[/tex] to [tex]\(1.75\)[/tex] is [tex]\(3.5 / 1.75 = 2\)[/tex].
- Ratio of [tex]\(7\)[/tex] to [tex]\(3.5\)[/tex] is [tex]\(7 / 3.5 = 2\)[/tex].
- Ratio of [tex]\(14\)[/tex] to [tex]\(7\)[/tex] is [tex]\(14 / 7 = 2\)[/tex].
Since the ratio is constant, this sequence is geometric.
### Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
An arithmetic sequence has a constant difference between consecutive terms:
- Difference between [tex]\(-10.8\)[/tex] and [tex]\(-12\)[/tex] is [tex]\(-10.8 - (-12) = 1.2\)[/tex].
- Difference between [tex]\(-9.6\)[/tex] and [tex]\(-10.8\)[/tex] is [tex]\(-9.6 - (-10.8) = 1.2\)[/tex].
- Difference between [tex]\(-8.4\)[/tex] and [tex]\(-9.6\)[/tex] is [tex]\(-8.4 - (-9.6) = 1.2\)[/tex].
Since the difference is constant, this sequence is arithmetic.
### Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
Check if it's arithmetic:
- Difference between [tex]\(1\)[/tex] and [tex]\(-1\)[/tex] is [tex]\(1 - (-1) = 2\)[/tex].
- Difference between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] is [tex]\(-1 - 1 = -2\)[/tex].
- Difference between [tex]\(1\)[/tex] and [tex]\(-1\)[/tex] is [tex]\(1 - (-1) = 2\)[/tex].
Check if it's geometric:
- Ratio of [tex]\(1\)[/tex] to [tex]\(-1\)[/tex] is [tex]\(1 / -1 = -1\)[/tex].
- Ratio of [tex]\(-1\)[/tex] to [tex]\(1\)[/tex] is [tex]\(-1 / 1 = -1\)[/tex].
- Ratio of [tex]\(1\)[/tex] to [tex]\(-1\)[/tex] is [tex]\(1 / -1 = -1\)[/tex].
Even though the ratios are constant, alternating signs generally suggest it's neither consistently growing nor shrinking.
Given the repeated pattern and switching signs, this sequence is neither arithmetic nor geometric.
### Summary
- [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]: Arithmetic
- [tex]\(1, 0, -1, 0, \ldots\)[/tex]: Neither
- [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]: Geometric
- [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]: Arithmetic
- [tex]\(-1, 1, -1, 1, \ldots\)[/tex]: Neither
I hope this helps! If you have any more questions, feel free to ask!
Thanks for taking the time to read Sort the sequences according to whether they are arithmetic geometric or neither 1 tex 98 3 94 1 89 9 85 7 ldots tex 2. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
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