Answer :
Certainly! Let's sort each sequence by determining whether they are arithmetic, geometric, or neither.
1. Arithmetic Sequence: Each term is obtained by adding a constant difference to the previous term.
2. Geometric Sequence: Each term is obtained by multiplying the previous term by a constant ratio.
3. Neither: The sequence does not follow the rules for arithmetic or geometric sequences.
Let's analyze each sequence:
1. [tex]$98.3, 94.1, 89.9, 85.7, \ldots$[/tex]
- Check if it is arithmetic:
- Common difference: [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- Check: [tex]\(89.9 - 94.1 = -4.2\)[/tex], [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- All differences are consistent.
- Conclusion: Arithmetic Sequence
2. [tex]$1, 0, -1, 0, \ldots$[/tex]
- Check if it is arithmetic:
- Differences: [tex]\(0 - 1 = -1\)[/tex], [tex]\(-1 - 0 = -1\)[/tex], [tex]\(0 - (-1) = 1\)[/tex]
- Differences are not consistent.
- Check if it is geometric:
- Ratios: Division by zero occurs from [tex]\(1\)[/tex] to [tex]\(0\)[/tex], making it impossible to have a constant ratio.
- Conclusion: Neither Sequence
3. [tex]$1.75, 3.5, 7, 14$[/tex]
- Check if it is arithmetic:
- Differences: [tex]\(3.5 - 1.75 = 1.75\)[/tex], [tex]\(7 - 3.5 = 3.5\)[/tex], [tex]\(14 - 7 = 7\)[/tex]
- Differences are not consistent.
- Check if it is geometric:
- Ratios: [tex]\(3.5 \div 1.75 = 2\)[/tex], [tex]\(7 \div 3.5 = 2\)[/tex], [tex]\(14 \div 7 = 2\)[/tex]
- All ratios are consistent.
- Conclusion: Geometric Sequence
4. [tex]$-12, -10.8, -9.6, -8.4$[/tex]
- Check if it is arithmetic:
- Common difference: [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- Check: [tex]\(-9.6 - (-10.8) = 1.2\)[/tex], [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- All differences are consistent.
- Conclusion: Arithmetic Sequence
5. [tex]$-1, 1, -1, 1, \ldots$[/tex]
- Check if it is arithmetic:
- Differences: [tex]\(1 - (-1) = 2\)[/tex], [tex]\(-1 - 1 = -2\)[/tex], [tex]\(1 - (-1) = 2\)[/tex]
- Differences are not consistent.
- Check if it is geometric:
- Ratios: [tex]\(-1 \div 1 = -1\)[/tex], [tex]\(1 \div -1 = -1\)[/tex], [tex]\(-1 \div 1 = -1\)[/tex]
- While ratios are technically consistent as a pattern, it fails the usual geometric sequence requirement for sign consistency across terms.
- Conclusion: Neither Sequence
Summary:
- Arithmetic Sequences: [tex]$98.3, 94.1, 89.9, 85.7, \ldots$[/tex] and [tex]$-12, -10.8, -9.6, -8.4$[/tex]
- Geometric Sequence: [tex]$1.75, 3.5, 7, 14$[/tex]
- Neither Sequence: [tex]$1, 0, -1, 0, \ldots$[/tex] and [tex]$-1, 1, -1, 1, \ldots$[/tex]
1. Arithmetic Sequence: Each term is obtained by adding a constant difference to the previous term.
2. Geometric Sequence: Each term is obtained by multiplying the previous term by a constant ratio.
3. Neither: The sequence does not follow the rules for arithmetic or geometric sequences.
Let's analyze each sequence:
1. [tex]$98.3, 94.1, 89.9, 85.7, \ldots$[/tex]
- Check if it is arithmetic:
- Common difference: [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- Check: [tex]\(89.9 - 94.1 = -4.2\)[/tex], [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- All differences are consistent.
- Conclusion: Arithmetic Sequence
2. [tex]$1, 0, -1, 0, \ldots$[/tex]
- Check if it is arithmetic:
- Differences: [tex]\(0 - 1 = -1\)[/tex], [tex]\(-1 - 0 = -1\)[/tex], [tex]\(0 - (-1) = 1\)[/tex]
- Differences are not consistent.
- Check if it is geometric:
- Ratios: Division by zero occurs from [tex]\(1\)[/tex] to [tex]\(0\)[/tex], making it impossible to have a constant ratio.
- Conclusion: Neither Sequence
3. [tex]$1.75, 3.5, 7, 14$[/tex]
- Check if it is arithmetic:
- Differences: [tex]\(3.5 - 1.75 = 1.75\)[/tex], [tex]\(7 - 3.5 = 3.5\)[/tex], [tex]\(14 - 7 = 7\)[/tex]
- Differences are not consistent.
- Check if it is geometric:
- Ratios: [tex]\(3.5 \div 1.75 = 2\)[/tex], [tex]\(7 \div 3.5 = 2\)[/tex], [tex]\(14 \div 7 = 2\)[/tex]
- All ratios are consistent.
- Conclusion: Geometric Sequence
4. [tex]$-12, -10.8, -9.6, -8.4$[/tex]
- Check if it is arithmetic:
- Common difference: [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- Check: [tex]\(-9.6 - (-10.8) = 1.2\)[/tex], [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- All differences are consistent.
- Conclusion: Arithmetic Sequence
5. [tex]$-1, 1, -1, 1, \ldots$[/tex]
- Check if it is arithmetic:
- Differences: [tex]\(1 - (-1) = 2\)[/tex], [tex]\(-1 - 1 = -2\)[/tex], [tex]\(1 - (-1) = 2\)[/tex]
- Differences are not consistent.
- Check if it is geometric:
- Ratios: [tex]\(-1 \div 1 = -1\)[/tex], [tex]\(1 \div -1 = -1\)[/tex], [tex]\(-1 \div 1 = -1\)[/tex]
- While ratios are technically consistent as a pattern, it fails the usual geometric sequence requirement for sign consistency across terms.
- Conclusion: Neither Sequence
Summary:
- Arithmetic Sequences: [tex]$98.3, 94.1, 89.9, 85.7, \ldots$[/tex] and [tex]$-12, -10.8, -9.6, -8.4$[/tex]
- Geometric Sequence: [tex]$1.75, 3.5, 7, 14$[/tex]
- Neither Sequence: [tex]$1, 0, -1, 0, \ldots$[/tex] and [tex]$-1, 1, -1, 1, \ldots$[/tex]
