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!
