Answer :
Sure, let's look at each sequence and determine whether they are arithmetic, geometric, or neither.
### Definitions:
- Arithmetic Sequence: A sequence of numbers in which the difference between consecutive terms is constant.
- Geometric Sequence: A sequence of numbers in which each term after the first is found by multiplying the previous term by a constant.
- Neither: Sequences that do not fit into either the arithmetic or geometric category.
### Sequence Analysis:
1. Sequence: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
- Common Difference:
[tex]\[
94.1 - 98.3 = -4.2 \\
89.9 - 94.1 = -4.2 \\
85.7 - 89.9 = -4.2
\][/tex]
- The difference between each consecutive term is consistent, indicating it is an arithmetic sequence.
2. Sequence: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
- This sequence alternates between 1, 0, and -1.
- There is no constant difference (for arithmetic) and no constant ratio (for geometric).
- Therefore, this sequence is neither arithmetic nor geometric.
3. Sequence: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Common Ratio:
[tex]\[
\frac{3.5}{1.75} = 2 \\
\frac{7}{3.5} = 2 \\
\frac{14}{7} = 2
\][/tex]
- The ratio between each consecutive term is consistent, indicating it is a geometric sequence.
4. Sequence: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Common Difference:
[tex]\[
-10.8 - (-12) = 1.2 \\
-9.6 - (-10.8) = 1.2 \\
-8.4 - (-9.6) = 1.2
\][/tex]
- The difference between each consecutive term is consistent, indicating it is an arithmetic sequence.
5. Sequence: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- This sequence alternates between -1 and 1.
- There is no constant difference (for arithmetic) and no constant ratio (for geometric).
- Therefore, this sequence is neither arithmetic nor geometric.
### Final Classification:
- Arithmetic Sequences: The 1st and the 4th sequences.
- Geometric Sequence: The 3rd sequence.
- Neither: The 2nd and 5th sequences.
So the sequences can be classified as:
- Arithmetic: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex] and [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Geometric: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Neither: [tex]\( 1, 0, -1, 0, \ldots \)[/tex] and [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
In terms of the question's list, the sequences are classified as:
- Arithmetic: [1, 4]
- Geometric: [3]
- Neither: [2, 5]
### Definitions:
- Arithmetic Sequence: A sequence of numbers in which the difference between consecutive terms is constant.
- Geometric Sequence: A sequence of numbers in which each term after the first is found by multiplying the previous term by a constant.
- Neither: Sequences that do not fit into either the arithmetic or geometric category.
### Sequence Analysis:
1. Sequence: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
- Common Difference:
[tex]\[
94.1 - 98.3 = -4.2 \\
89.9 - 94.1 = -4.2 \\
85.7 - 89.9 = -4.2
\][/tex]
- The difference between each consecutive term is consistent, indicating it is an arithmetic sequence.
2. Sequence: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
- This sequence alternates between 1, 0, and -1.
- There is no constant difference (for arithmetic) and no constant ratio (for geometric).
- Therefore, this sequence is neither arithmetic nor geometric.
3. Sequence: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Common Ratio:
[tex]\[
\frac{3.5}{1.75} = 2 \\
\frac{7}{3.5} = 2 \\
\frac{14}{7} = 2
\][/tex]
- The ratio between each consecutive term is consistent, indicating it is a geometric sequence.
4. Sequence: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Common Difference:
[tex]\[
-10.8 - (-12) = 1.2 \\
-9.6 - (-10.8) = 1.2 \\
-8.4 - (-9.6) = 1.2
\][/tex]
- The difference between each consecutive term is consistent, indicating it is an arithmetic sequence.
5. Sequence: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- This sequence alternates between -1 and 1.
- There is no constant difference (for arithmetic) and no constant ratio (for geometric).
- Therefore, this sequence is neither arithmetic nor geometric.
### Final Classification:
- Arithmetic Sequences: The 1st and the 4th sequences.
- Geometric Sequence: The 3rd sequence.
- Neither: The 2nd and 5th sequences.
So the sequences can be classified as:
- Arithmetic: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex] and [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Geometric: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Neither: [tex]\( 1, 0, -1, 0, \ldots \)[/tex] and [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
In terms of the question's list, the sequences are classified as:
- Arithmetic: [1, 4]
- Geometric: [3]
- Neither: [2, 5]
