Answer :
To determine the type of each sequence among arithmetic, geometric, or neither, we will analyze the characteristics of each sequence individually:
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
An arithmetic sequence has a constant difference between consecutive terms. Let's check this:
- Difference between 94.1 and 98.3 is: [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- Difference between 89.9 and 94.1 is: [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- Difference between 85.7 and 89.9 is: [tex]\(85.7 - 89.9 = -4.2\)[/tex]
Since the differences are the same, this sequence is arithmetic.
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
For it to be arithmetic, it must have a constant difference between terms. For it to be geometric, it must have a constant ratio. Let's check:
- The differences are not constant: [tex]\(0 - 1 = -1\)[/tex], [tex]\(-1 - 0 = -1\)[/tex], [tex]\(0 - (-1) = 1\)[/tex]
- The ratios are also not constant: [tex]\(0/1 = 0\)[/tex], [tex]\((-1)/0\)[/tex] is undefined, [tex]\(0/(-1) = 0\)[/tex]
This sequence does not have a constant difference or ratio. Therefore, it is neither.
3. Sequence 3: [tex]\(1.75, 3.5, 7, 14\)[/tex]
A geometric sequence has a constant ratio between terms. Let's analyze:
- Ratio between 3.5 and 1.75 is: [tex]\(3.5 / 1.75 = 2\)[/tex]
- Ratio between 7 and 3.5 is: [tex]\(7 / 3.5 = 2\)[/tex]
- Ratio between 14 and 7 is: [tex]\(14 / 7 = 2\)[/tex]
As the ratios are constant, this sequence is geometric.
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
Checking for constant differences to determine if it is arithmetic:
- 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]
The differences are the same, indicating this sequence is arithmetic.
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
To determine if this is arithmetic or geometric:
- Arithmetic check: The differences are [tex]\((1 - (-1) = 2)\)[/tex] and [tex]\((-1 - 1 = -2)\)[/tex], which are not constant.
- Geometric check: The ratios are [tex]\((1 / -1 = -1)\)[/tex] and [tex]\((-1 / 1 = -1)\)[/tex], alternating signs with no constant behavior.
This sequence shows an alternating pattern without a consistent difference or ratio, so it is neither.
In summary:
- Sequence 1 is arithmetic.
- Sequence 2 is neither arithmetic nor geometric.
- Sequence 3 is geometric.
- Sequence 4 is arithmetic.
- Sequence 5 is neither arithmetic nor geometric.
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
An arithmetic sequence has a constant difference between consecutive terms. Let's check this:
- Difference between 94.1 and 98.3 is: [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- Difference between 89.9 and 94.1 is: [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- Difference between 85.7 and 89.9 is: [tex]\(85.7 - 89.9 = -4.2\)[/tex]
Since the differences are the same, this sequence is arithmetic.
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
For it to be arithmetic, it must have a constant difference between terms. For it to be geometric, it must have a constant ratio. Let's check:
- The differences are not constant: [tex]\(0 - 1 = -1\)[/tex], [tex]\(-1 - 0 = -1\)[/tex], [tex]\(0 - (-1) = 1\)[/tex]
- The ratios are also not constant: [tex]\(0/1 = 0\)[/tex], [tex]\((-1)/0\)[/tex] is undefined, [tex]\(0/(-1) = 0\)[/tex]
This sequence does not have a constant difference or ratio. Therefore, it is neither.
3. Sequence 3: [tex]\(1.75, 3.5, 7, 14\)[/tex]
A geometric sequence has a constant ratio between terms. Let's analyze:
- Ratio between 3.5 and 1.75 is: [tex]\(3.5 / 1.75 = 2\)[/tex]
- Ratio between 7 and 3.5 is: [tex]\(7 / 3.5 = 2\)[/tex]
- Ratio between 14 and 7 is: [tex]\(14 / 7 = 2\)[/tex]
As the ratios are constant, this sequence is geometric.
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
Checking for constant differences to determine if it is arithmetic:
- 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]
The differences are the same, indicating this sequence is arithmetic.
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
To determine if this is arithmetic or geometric:
- Arithmetic check: The differences are [tex]\((1 - (-1) = 2)\)[/tex] and [tex]\((-1 - 1 = -2)\)[/tex], which are not constant.
- Geometric check: The ratios are [tex]\((1 / -1 = -1)\)[/tex] and [tex]\((-1 / 1 = -1)\)[/tex], alternating signs with no constant behavior.
This sequence shows an alternating pattern without a consistent difference or ratio, so it is neither.
In summary:
- Sequence 1 is arithmetic.
- Sequence 2 is neither arithmetic nor geometric.
- Sequence 3 is geometric.
- Sequence 4 is arithmetic.
- Sequence 5 is neither arithmetic nor geometric.
