Answer :
Sure, let's analyze each sequence step-by-step to determine whether they are arithmetic, geometric, or neither.
### Definitions:
- Arithmetic Sequence: A sequence of numbers is arithmetic if the difference between consecutive terms is constant. This difference is called the common difference.
- Geometric Sequence: A sequence of numbers is geometric if the ratio between consecutive terms is constant. This ratio is called the common ratio.
Now we'll apply these definitions to each sequence:
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7\)[/tex]
- First, find the difference 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]
- The difference is constant ([tex]\(-4.2\)[/tex]), so this is an arithmetic sequence.
2. Sequence: [tex]\(1, 0, -1, 0\)[/tex]
- First, find the difference between consecutive terms:
[tex]\(0 - 1 = -1\)[/tex]
[tex]\(-1 - 0 = -1\)[/tex]
[tex]\(0 - (-1) = 1\)[/tex]
- The difference is not constant, so it is not arithmetic.
- Next, find the ratio between consecutive terms:
[tex]\(0/1 = 0\)[/tex]
[tex]\(-1/0\)[/tex] (Undefined)
- The ratio is not constant, so it is not geometric.
- Therefore, this sequence is neither.
3. Sequence: [tex]\(1.75, 35, 7, 14\)[/tex]
- First, find the difference between consecutive terms:
[tex]\(35 - 1.75 = 33.25\)[/tex]
[tex]\(7 - 35 = -28\)[/tex]
[tex]\(14 - 7 = 7\)[/tex]
- The difference is not constant, so it is not arithmetic.
- Next, find the ratio between consecutive terms:
[tex]\(35 / 1.75 = 20\)[/tex]
[tex]\(7 / 35 = 1/5\)[/tex]
[tex]\(14 / 7 = 2\)[/tex]
- The ratio is not constant, so it is not geometric.
- Therefore, this sequence is neither.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- First, find the difference 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]
- The difference is constant ([tex]\(1.2\)[/tex]), so this is an arithmetic sequence.
5. Sequence: [tex]\(-1, 1, -1, 1\)[/tex]
- First, find the difference between consecutive terms:
[tex]\(1 - (-1) = 2\)[/tex]
[tex]\(-1 - 1 = -2\)[/tex]
[tex]\(1 - (-1) = 2\)[/tex]
- The difference is not constant, so it is not arithmetic.
- Next, find the ratio between consecutive terms:
[tex]\(1 / -1 = -1\)[/tex]
[tex]\(-1 / 1 = -1\)[/tex]
[tex]\(1 / -1 = -1\)[/tex]
- The ratio is constant ([tex]\(-1\)[/tex]), so this is a geometric sequence.
### Summary:
- 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]\(1.75, 35, 7, 14\)[/tex]
I hope this helps! If you have any more questions, feel free to ask.
### Definitions:
- Arithmetic Sequence: A sequence of numbers is arithmetic if the difference between consecutive terms is constant. This difference is called the common difference.
- Geometric Sequence: A sequence of numbers is geometric if the ratio between consecutive terms is constant. This ratio is called the common ratio.
Now we'll apply these definitions to each sequence:
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7\)[/tex]
- First, find the difference 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]
- The difference is constant ([tex]\(-4.2\)[/tex]), so this is an arithmetic sequence.
2. Sequence: [tex]\(1, 0, -1, 0\)[/tex]
- First, find the difference between consecutive terms:
[tex]\(0 - 1 = -1\)[/tex]
[tex]\(-1 - 0 = -1\)[/tex]
[tex]\(0 - (-1) = 1\)[/tex]
- The difference is not constant, so it is not arithmetic.
- Next, find the ratio between consecutive terms:
[tex]\(0/1 = 0\)[/tex]
[tex]\(-1/0\)[/tex] (Undefined)
- The ratio is not constant, so it is not geometric.
- Therefore, this sequence is neither.
3. Sequence: [tex]\(1.75, 35, 7, 14\)[/tex]
- First, find the difference between consecutive terms:
[tex]\(35 - 1.75 = 33.25\)[/tex]
[tex]\(7 - 35 = -28\)[/tex]
[tex]\(14 - 7 = 7\)[/tex]
- The difference is not constant, so it is not arithmetic.
- Next, find the ratio between consecutive terms:
[tex]\(35 / 1.75 = 20\)[/tex]
[tex]\(7 / 35 = 1/5\)[/tex]
[tex]\(14 / 7 = 2\)[/tex]
- The ratio is not constant, so it is not geometric.
- Therefore, this sequence is neither.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- First, find the difference 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]
- The difference is constant ([tex]\(1.2\)[/tex]), so this is an arithmetic sequence.
5. Sequence: [tex]\(-1, 1, -1, 1\)[/tex]
- First, find the difference between consecutive terms:
[tex]\(1 - (-1) = 2\)[/tex]
[tex]\(-1 - 1 = -2\)[/tex]
[tex]\(1 - (-1) = 2\)[/tex]
- The difference is not constant, so it is not arithmetic.
- Next, find the ratio between consecutive terms:
[tex]\(1 / -1 = -1\)[/tex]
[tex]\(-1 / 1 = -1\)[/tex]
[tex]\(1 / -1 = -1\)[/tex]
- The ratio is constant ([tex]\(-1\)[/tex]), so this is a geometric sequence.
### Summary:
- 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]\(1.75, 35, 7, 14\)[/tex]
I hope this helps! If you have any more questions, feel free to ask.
