Answer :
Let's sort each sequence based on whether they are arithmetic, geometric, or neither. We'll examine the differences between consecutive terms for arithmetic sequences and the ratios for geometric sequences.
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Check for Arithmetic Sequence:
- 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]
- The differences are consistent. Therefore, this sequence is arithmetic with a common difference of [tex]\(-4.2\)[/tex].
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Check for Arithmetic Sequence:
- Calculate the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- The differences are not consistent.
- Check for Geometric Sequence:
- You can't calculate the ratio with terms equal to zero, making it inconsistent.
- Since it doesn't fit either pattern, this sequence is neither.
3. Sequence: [tex]\(1.75, 35, 7, 14\)[/tex]
- Check for Arithmetic Sequence:
- Calculate the differences:
- [tex]\(35 - 1.75 = 33.25\)[/tex]
- [tex]\(7 - 35 = -28\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
- The differences are not consistent.
- Check for Geometric Sequence:
- Calculate the ratios:
- [tex]\(35 / 1.75 = 20\)[/tex]
- [tex]\(7 / 35 = 0.2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- The ratios are not consistent.
- This sequence is neither.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check for Arithmetic Sequence:
- 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]
- The differences are consistent. Therefore, this sequence is arithmetic with a common difference of [tex]\(1.2\)[/tex].
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Check for Arithmetic Sequence:
- Calculate the differences:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- The differences are not consistent.
- Check for Geometric Sequence:
- Calculate the ratios:
- [tex]\(1 / -1 = -1\)[/tex]
- [tex]\(-1 / 1 = -1\)[/tex]
- [tex]\(1 / -1 = -1\)[/tex]
- The ratios are consistent (-1 each time). Therefore, this sequence is geometric with a common ratio of [tex]\(-1\)[/tex].
Summary:
- Arithmetic: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex], [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Geometric: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Neither: [tex]\(1, 0, -1, 0, \ldots\)[/tex], [tex]\(1.75, 35, 7, 14\)[/tex]
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- Check for Arithmetic Sequence:
- 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]
- The differences are consistent. Therefore, this sequence is arithmetic with a common difference of [tex]\(-4.2\)[/tex].
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Check for Arithmetic Sequence:
- Calculate the differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- The differences are not consistent.
- Check for Geometric Sequence:
- You can't calculate the ratio with terms equal to zero, making it inconsistent.
- Since it doesn't fit either pattern, this sequence is neither.
3. Sequence: [tex]\(1.75, 35, 7, 14\)[/tex]
- Check for Arithmetic Sequence:
- Calculate the differences:
- [tex]\(35 - 1.75 = 33.25\)[/tex]
- [tex]\(7 - 35 = -28\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
- The differences are not consistent.
- Check for Geometric Sequence:
- Calculate the ratios:
- [tex]\(35 / 1.75 = 20\)[/tex]
- [tex]\(7 / 35 = 0.2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- The ratios are not consistent.
- This sequence is neither.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check for Arithmetic Sequence:
- 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]
- The differences are consistent. Therefore, this sequence is arithmetic with a common difference of [tex]\(1.2\)[/tex].
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Check for Arithmetic Sequence:
- Calculate the differences:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- The differences are not consistent.
- Check for Geometric Sequence:
- Calculate the ratios:
- [tex]\(1 / -1 = -1\)[/tex]
- [tex]\(-1 / 1 = -1\)[/tex]
- [tex]\(1 / -1 = -1\)[/tex]
- The ratios are consistent (-1 each time). Therefore, this sequence is geometric with a common ratio of [tex]\(-1\)[/tex].
Summary:
- Arithmetic: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex], [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Geometric: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Neither: [tex]\(1, 0, -1, 0, \ldots\)[/tex], [tex]\(1.75, 35, 7, 14\)[/tex]
