Answer :
Let's classify each sequence step-by-step to determine whether they are arithmetic, geometric, or neither.
### Sequence 1: [tex]\( 98.3, 94.1, 89.9, 85.7 \)[/tex]
Let's find the common 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]
Since the difference between each term is constant ([tex]\(-4.2\)[/tex]), this sequence is arithmetic.
### Sequence 2: [tex]\( 1, 0, -1, 0, 1, -1, \ldots \)[/tex]
Let's check the differences between consecutive terms:
- [tex]\( 0 - 1 = -1 \)[/tex]
- [tex]\(-1 - 0 = -1 \)[/tex]
- [tex]\( 0 - (-1) = 1 \)[/tex]
- [tex]\( 1 - 0 = 1 \)[/tex]
- [tex]\(-1 - 1 = -2 \)[/tex]
The differences are not constant and do not form a consistent pattern. Also, checking the ratios:
- [tex]\( 0 / 1 = 0 \)[/tex]
- [tex]\(-1 / 0\)[/tex] is undefined (division by zero)
- [tex]\( 0 / -1 = 0 \)[/tex]
- [tex]\( 1 / 0\)[/tex] is undefined
This sequence is neither arithmetic nor geometric.
### Sequence 3: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
Let's find the common ratio:
- [tex]\( 3.5 / 1.75 = 2 \)[/tex]
- [tex]\( 7 / 3.5 = 2 \)[/tex]
- [tex]\( 14 / 7 = 2 \)[/tex]
Since the ratio between each term is constant ([tex]\(2\)[/tex]), this sequence is geometric.
### Sequence 4: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
Let's find the common 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]
Since the difference between each term is constant ([tex]\(1.2\)[/tex]), this sequence is arithmetic.
### Sequence 5: [tex]\( -1, 1, -1, 1, -1, \ldots \)[/tex]
The sequence alternates between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex]. Let's check the differences:
- [tex]\( 1 - (-1) = 2 \)[/tex]
- [tex]\( -1 - 1 = -2 \)[/tex]
- [tex]\( 1 - (-1) = 2 \)[/tex]
- [tex]\( -1 - 1 = -2 \)[/tex]
The differences alternate between positive and negative values. Examining the sequence more closely:
- [tex]\( 1 / (-1) = -1 \)[/tex]
- [tex]\( -1 / 1 = -1 \)[/tex]
The ratios are consistent but switch signs. Given the alternating nature, it doesn't rigidly conform to arithmetic or geometric properties in a standard sense.
This sequence is neither arithmetic nor geometric.
### Summary of Classification:
Arithmetic:
- [tex]\( 98.3, 94.1, 89.9, 85.7 \)[/tex]
- [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
Geometric:
- [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
Neither:
- [tex]\( 1, 0, -1, 0, 1, -1, \ldots \)[/tex]
- [tex]\( -1, 1, -1, 1, -1, \ldots \)[/tex]
Therefore, the sequences sorted according to their types are:
- Arithmetic:
- [tex]\( 98.3, 94.1, 89.9, 85.7 \)[/tex]
- [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Geometric:
- [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Neither:
- [tex]\( 1, 0, -1, 0, 1, -1, \ldots \)[/tex]
- [tex]\( -1, 1, -1, 1, -1, \ldots \)[/tex]
### Sequence 1: [tex]\( 98.3, 94.1, 89.9, 85.7 \)[/tex]
Let's find the common 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]
Since the difference between each term is constant ([tex]\(-4.2\)[/tex]), this sequence is arithmetic.
### Sequence 2: [tex]\( 1, 0, -1, 0, 1, -1, \ldots \)[/tex]
Let's check the differences between consecutive terms:
- [tex]\( 0 - 1 = -1 \)[/tex]
- [tex]\(-1 - 0 = -1 \)[/tex]
- [tex]\( 0 - (-1) = 1 \)[/tex]
- [tex]\( 1 - 0 = 1 \)[/tex]
- [tex]\(-1 - 1 = -2 \)[/tex]
The differences are not constant and do not form a consistent pattern. Also, checking the ratios:
- [tex]\( 0 / 1 = 0 \)[/tex]
- [tex]\(-1 / 0\)[/tex] is undefined (division by zero)
- [tex]\( 0 / -1 = 0 \)[/tex]
- [tex]\( 1 / 0\)[/tex] is undefined
This sequence is neither arithmetic nor geometric.
### Sequence 3: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
Let's find the common ratio:
- [tex]\( 3.5 / 1.75 = 2 \)[/tex]
- [tex]\( 7 / 3.5 = 2 \)[/tex]
- [tex]\( 14 / 7 = 2 \)[/tex]
Since the ratio between each term is constant ([tex]\(2\)[/tex]), this sequence is geometric.
### Sequence 4: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
Let's find the common 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]
Since the difference between each term is constant ([tex]\(1.2\)[/tex]), this sequence is arithmetic.
### Sequence 5: [tex]\( -1, 1, -1, 1, -1, \ldots \)[/tex]
The sequence alternates between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex]. Let's check the differences:
- [tex]\( 1 - (-1) = 2 \)[/tex]
- [tex]\( -1 - 1 = -2 \)[/tex]
- [tex]\( 1 - (-1) = 2 \)[/tex]
- [tex]\( -1 - 1 = -2 \)[/tex]
The differences alternate between positive and negative values. Examining the sequence more closely:
- [tex]\( 1 / (-1) = -1 \)[/tex]
- [tex]\( -1 / 1 = -1 \)[/tex]
The ratios are consistent but switch signs. Given the alternating nature, it doesn't rigidly conform to arithmetic or geometric properties in a standard sense.
This sequence is neither arithmetic nor geometric.
### Summary of Classification:
Arithmetic:
- [tex]\( 98.3, 94.1, 89.9, 85.7 \)[/tex]
- [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
Geometric:
- [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
Neither:
- [tex]\( 1, 0, -1, 0, 1, -1, \ldots \)[/tex]
- [tex]\( -1, 1, -1, 1, -1, \ldots \)[/tex]
Therefore, the sequences sorted according to their types are:
- Arithmetic:
- [tex]\( 98.3, 94.1, 89.9, 85.7 \)[/tex]
- [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Geometric:
- [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Neither:
- [tex]\( 1, 0, -1, 0, 1, -1, \ldots \)[/tex]
- [tex]\( -1, 1, -1, 1, -1, \ldots \)[/tex]
