Answer :
To determine whether each sequence is arithmetic, geometric, or neither, let's go through each one and see how they fit the definitions:
1. Arithmetic Sequence: This is a sequence where the difference between consecutive terms is constant. This difference is called the common difference.
2. Geometric Sequence: This is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Let's evaluate each sequence:
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7\)[/tex]
- Check for arithmetic: Calculate the differences between consecutive terms:
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- All differences are the same, so this sequence is an Arithmetic Sequence.
2. Sequence: [tex]\(1, 0, -1, 0\)[/tex]
- Check for arithmetic: Differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- Differences are not consistent.
- It is neither arithmetic nor geometric because it doesn't follow a specific ratio either.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Check for geometric: Ratios:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- All ratios are the same (2), so this sequence is a Geometric Sequence.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check for arithmetic: 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 the same, so this sequence is an Arithmetic Sequence.
5. Sequence: [tex]\(-1, 1, -1, 1\)[/tex]
- Check for neither: Alternating pattern with no consistent difference or ratio.
- It is neither an arithmetic nor a geometric sequence.
After analyzing these sequences:
- Arithmetic Sequences: [tex]\(98.3, 94.1, 89.9, 85.7\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Geometric Sequences: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Neither: [tex]\(1, 0, -1, 0\)[/tex] and [tex]\(-1, 1, -1, 1\)[/tex]
1. Arithmetic Sequence: This is a sequence where the difference between consecutive terms is constant. This difference is called the common difference.
2. Geometric Sequence: This is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Let's evaluate each sequence:
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7\)[/tex]
- Check for arithmetic: Calculate the differences between consecutive terms:
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- All differences are the same, so this sequence is an Arithmetic Sequence.
2. Sequence: [tex]\(1, 0, -1, 0\)[/tex]
- Check for arithmetic: Differences:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- Differences are not consistent.
- It is neither arithmetic nor geometric because it doesn't follow a specific ratio either.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Check for geometric: Ratios:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- All ratios are the same (2), so this sequence is a Geometric Sequence.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check for arithmetic: 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 the same, so this sequence is an Arithmetic Sequence.
5. Sequence: [tex]\(-1, 1, -1, 1\)[/tex]
- Check for neither: Alternating pattern with no consistent difference or ratio.
- It is neither an arithmetic nor a geometric sequence.
After analyzing these sequences:
- Arithmetic Sequences: [tex]\(98.3, 94.1, 89.9, 85.7\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Geometric Sequences: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Neither: [tex]\(1, 0, -1, 0\)[/tex] and [tex]\(-1, 1, -1, 1\)[/tex]
