Answer :
Sure! Let's analyze each of the sequences step-by-step to determine their types:
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To check if it's an arithmetic sequence, we look for a constant difference between consecutive terms.
- Calculate the differences:
- [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 all differences are constant ([tex]\(-4.2\)[/tex]), this is an arithmetic sequence.
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- For arithmetic sequences, each term should have a constant difference.
- Calculate the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are not constant, so this is neither arithmetic nor geometric.
3. Sequence 3: [tex]\(1.75, 3.5, 7.14\)[/tex]
- Check for a constant ratio for a geometric sequence.
- Calculate the ratios:
- [tex]\(\frac{3.5}{1.75} \approx 2\)[/tex]
- [tex]\(\frac{7.14}{3.5} \approx 2.04\)[/tex]
Since the ratios are not exactly the same, this is neither geometric nor arithmetic.
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check for constant differences for arithmetic sequences.
- Calculate the differences:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
All differences are constant ([tex]\(1.2\)[/tex]), so this is an arithmetic sequence.
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- For a sequence to be arithmetic, it must have a constant difference.
- Calculate the differences:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
The differences are not constant, so this is neither arithmetic nor geometric.
In conclusion, based on our analysis, the sequences are classified as follows:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Neither
- Sequence 4: Arithmetic
- Sequence 5: Neither
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To check if it's an arithmetic sequence, we look for a constant difference between consecutive terms.
- Calculate the differences:
- [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 all differences are constant ([tex]\(-4.2\)[/tex]), this is an arithmetic sequence.
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- For arithmetic sequences, each term should have a constant difference.
- Calculate the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
The differences are not constant, so this is neither arithmetic nor geometric.
3. Sequence 3: [tex]\(1.75, 3.5, 7.14\)[/tex]
- Check for a constant ratio for a geometric sequence.
- Calculate the ratios:
- [tex]\(\frac{3.5}{1.75} \approx 2\)[/tex]
- [tex]\(\frac{7.14}{3.5} \approx 2.04\)[/tex]
Since the ratios are not exactly the same, this is neither geometric nor arithmetic.
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check for constant differences for arithmetic sequences.
- Calculate the differences:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
All differences are constant ([tex]\(1.2\)[/tex]), so this is an arithmetic sequence.
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- For a sequence to be arithmetic, it must have a constant difference.
- Calculate the differences:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
The differences are not constant, so this is neither arithmetic nor geometric.
In conclusion, based on our analysis, the sequences are classified as follows:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Neither
- Sequence 4: Arithmetic
- Sequence 5: Neither
