Answer :
Sure! Let's examine each sequence one by one and determine whether they are arithmetic, geometric, or neither.
### Sequence 1: 98.3, 94.1, 89.9, 85.7, ...
To check if this is an arithmetic sequence, we need to see if the difference between consecutive terms is constant.
- [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 common difference is [tex]\(-4.2\)[/tex], which is constant. Hence, Sequence 1 is arithmetic.
### Sequence 2: 1, 0, -1, 0, ...
To check if this is an arithmetic sequence:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The difference is not constant.
To check if this is a geometric sequence:
- [tex]\(0 / 1 = 0\)[/tex]
- [tex]\(-1 / 0\)[/tex] is undefined
Since the division by zero is not possible, and the ratios are not consistent, this sequence is neither arithmetic nor geometric. Sequence 2 is neither.
### Sequence 3: 1.75, 3.5, 7, 14
To check if this is an arithmetic sequence:
- [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- [tex]\(7 - 3.5 = 3.5\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
The common difference is not constant.
To check if this is a geometric sequence:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
The common ratio is 2, which is constant. Hence, Sequence 3 is geometric.
### Sequence 4: -12, -10.8, -9.6, -8.4
To check if this is an arithmetic sequence:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
The common difference is [tex]\(1.2\)[/tex], which is constant. Hence, Sequence 4 is arithmetic.
### Sequence 5: -1, 1, -1, 1, ...
To check if this is an arithmetic sequence:
- [tex]\( 1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
The difference is not constant.
To check if this is a geometric sequence:
- [tex]\(1 / (-1) = -1\)[/tex]
- [tex]\(-1 / 1 = -1\)[/tex]
- [tex]\(1 / (-1) = -1\)[/tex]
The common ratio is [tex]\(-1\)[/tex], which is constant. Hence, Sequence 5 is geometric.
### Summary:
1. [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] is arithmetic.
2. [tex]\(1, 0, -1, 0, \ldots\)[/tex] is neither.
3. [tex]\(1.75, 3.5, 7, 14\)[/tex] is geometric.
4. [tex]\(-12, -10.8, -9.6, -8.4\)[/tex] is arithmetic.
5. [tex]\(-1, 1, -1, 1, \ldots\)[/tex] is geometric.
### Sequence 1: 98.3, 94.1, 89.9, 85.7, ...
To check if this is an arithmetic sequence, we need to see if the difference between consecutive terms is constant.
- [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 common difference is [tex]\(-4.2\)[/tex], which is constant. Hence, Sequence 1 is arithmetic.
### Sequence 2: 1, 0, -1, 0, ...
To check if this is an arithmetic sequence:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The difference is not constant.
To check if this is a geometric sequence:
- [tex]\(0 / 1 = 0\)[/tex]
- [tex]\(-1 / 0\)[/tex] is undefined
Since the division by zero is not possible, and the ratios are not consistent, this sequence is neither arithmetic nor geometric. Sequence 2 is neither.
### Sequence 3: 1.75, 3.5, 7, 14
To check if this is an arithmetic sequence:
- [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- [tex]\(7 - 3.5 = 3.5\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
The common difference is not constant.
To check if this is a geometric sequence:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
The common ratio is 2, which is constant. Hence, Sequence 3 is geometric.
### Sequence 4: -12, -10.8, -9.6, -8.4
To check if this is an arithmetic sequence:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
The common difference is [tex]\(1.2\)[/tex], which is constant. Hence, Sequence 4 is arithmetic.
### Sequence 5: -1, 1, -1, 1, ...
To check if this is an arithmetic sequence:
- [tex]\( 1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
The difference is not constant.
To check if this is a geometric sequence:
- [tex]\(1 / (-1) = -1\)[/tex]
- [tex]\(-1 / 1 = -1\)[/tex]
- [tex]\(1 / (-1) = -1\)[/tex]
The common ratio is [tex]\(-1\)[/tex], which is constant. Hence, Sequence 5 is geometric.
### Summary:
1. [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] is arithmetic.
2. [tex]\(1, 0, -1, 0, \ldots\)[/tex] is neither.
3. [tex]\(1.75, 3.5, 7, 14\)[/tex] is geometric.
4. [tex]\(-12, -10.8, -9.6, -8.4\)[/tex] is arithmetic.
5. [tex]\(-1, 1, -1, 1, \ldots\)[/tex] is geometric.
