Answer :
To determine which sequences are arithmetic sequences, we need to check if each sequence has a constant difference between consecutive terms. An arithmetic sequence is one where the difference [tex]\(d\)[/tex] between consecutive terms [tex]\(a_n\)[/tex] and [tex]\(a_{n-1}\)[/tex] (i.e., [tex]\(a_n - a_{n-1}\)[/tex]) is the same throughout the sequence.
Let's examine each sequence step-by-step:
1. Sequence: [tex]\(-8.6, -5.0, -1.4, 2.2, 5.8, \ldots\)[/tex]
- Differences:
- [tex]\(-5.0 - (-8.6) = 3.6\)[/tex]
- [tex]\(-1.4 - (-5.0) = 3.6\)[/tex]
- [tex]\(2.2 - (-1.4) = 3.6\)[/tex]
- [tex]\(5.8 - 2.2 = 3.6\)[/tex]
- Since all the differences are [tex]\(3.6\)[/tex], this sequence is not an arithmetic sequence as the differences are not consistent.
2. Sequence: [tex]\(2, -2.2, 2.42, -2.662, 2.9282, \ldots\)[/tex]
- Differences:
- [tex]\(-2.2 - 2 = -4.2\)[/tex]
- [tex]\(2.42 - (-2.2) = 4.62\)[/tex]
- [tex]\(-2.662 - 2.42 = -5.082\)[/tex]
- [tex]\(2.9282 - (-2.662) = 5.5902\)[/tex]
- The differences are not consistent, so this sequence is not an arithmetic sequence.
3. Sequence: [tex]\(5, 1, -3, -7, -11, \ldots\)[/tex]
- Differences:
- [tex]\(1 - 5 = -4\)[/tex]
- [tex]\(-3 - 1 = -4\)[/tex]
- [tex]\(-7 - (-3) = -4\)[/tex]
- [tex]\(-11 - (-7) = -4\)[/tex]
- All the differences are [tex]\(-4\)[/tex], making this sequence an arithmetic sequence.
4. Sequence: [tex]\(-3, 3, 9, 15, 21, \ldots\)[/tex]
- Differences:
- [tex]\(3 - (-3) = 6\)[/tex]
- [tex]\(9 - 3 = 6\)[/tex]
- [tex]\(15 - 9 = 6\)[/tex]
- [tex]\(21 - 15 = 6\)[/tex]
- All the differences are [tex]\(6\)[/tex], making this sequence an arithmetic sequence.
5. Sequence: [tex]\(-6.2, -3.1, -1.55, -0.775, -0.3875, \ldots\)[/tex]
- Differences:
- [tex]\(-3.1 - (-6.2) = 3.1\)[/tex]
- [tex]\(-1.55 - (-3.1) = 1.55\)[/tex]
- [tex]\(-0.775 - (-1.55) = 0.775\)[/tex]
- [tex]\(-0.3875 - (-0.775) = 0.3875\)[/tex]
- The differences are not consistent, so this sequence is not an arithmetic sequence.
Thus, the sequences which are arithmetic are:
1. [tex]\(5, 1, -3, -7, -11, \ldots\)[/tex]
2. [tex]\(-3, 3, 9, 15, 21, \ldots\)[/tex]
These are the arithmetic sequences from the given options.
Let's examine each sequence step-by-step:
1. Sequence: [tex]\(-8.6, -5.0, -1.4, 2.2, 5.8, \ldots\)[/tex]
- Differences:
- [tex]\(-5.0 - (-8.6) = 3.6\)[/tex]
- [tex]\(-1.4 - (-5.0) = 3.6\)[/tex]
- [tex]\(2.2 - (-1.4) = 3.6\)[/tex]
- [tex]\(5.8 - 2.2 = 3.6\)[/tex]
- Since all the differences are [tex]\(3.6\)[/tex], this sequence is not an arithmetic sequence as the differences are not consistent.
2. Sequence: [tex]\(2, -2.2, 2.42, -2.662, 2.9282, \ldots\)[/tex]
- Differences:
- [tex]\(-2.2 - 2 = -4.2\)[/tex]
- [tex]\(2.42 - (-2.2) = 4.62\)[/tex]
- [tex]\(-2.662 - 2.42 = -5.082\)[/tex]
- [tex]\(2.9282 - (-2.662) = 5.5902\)[/tex]
- The differences are not consistent, so this sequence is not an arithmetic sequence.
3. Sequence: [tex]\(5, 1, -3, -7, -11, \ldots\)[/tex]
- Differences:
- [tex]\(1 - 5 = -4\)[/tex]
- [tex]\(-3 - 1 = -4\)[/tex]
- [tex]\(-7 - (-3) = -4\)[/tex]
- [tex]\(-11 - (-7) = -4\)[/tex]
- All the differences are [tex]\(-4\)[/tex], making this sequence an arithmetic sequence.
4. Sequence: [tex]\(-3, 3, 9, 15, 21, \ldots\)[/tex]
- Differences:
- [tex]\(3 - (-3) = 6\)[/tex]
- [tex]\(9 - 3 = 6\)[/tex]
- [tex]\(15 - 9 = 6\)[/tex]
- [tex]\(21 - 15 = 6\)[/tex]
- All the differences are [tex]\(6\)[/tex], making this sequence an arithmetic sequence.
5. Sequence: [tex]\(-6.2, -3.1, -1.55, -0.775, -0.3875, \ldots\)[/tex]
- Differences:
- [tex]\(-3.1 - (-6.2) = 3.1\)[/tex]
- [tex]\(-1.55 - (-3.1) = 1.55\)[/tex]
- [tex]\(-0.775 - (-1.55) = 0.775\)[/tex]
- [tex]\(-0.3875 - (-0.775) = 0.3875\)[/tex]
- The differences are not consistent, so this sequence is not an arithmetic sequence.
Thus, the sequences which are arithmetic are:
1. [tex]\(5, 1, -3, -7, -11, \ldots\)[/tex]
2. [tex]\(-3, 3, 9, 15, 21, \ldots\)[/tex]
These are the arithmetic sequences from the given options.
