Answer :
To determine which sequences are arithmetic, let's recall the definition of an arithmetic sequence. In an arithmetic sequence, the difference between consecutive terms is constant. This difference is called the common difference.
Let's carefully analyze each of the given sequences to identify if they are arithmetic or not.
1. Sequence: [tex]\(-8.6, -5.0, -1.4, 2.2, 5.8, \ldots\)[/tex]
- Compute differences between consecutive terms:
- [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 the differences are all the same (3.6), this sequence is arithmetic.
2. Sequence: [tex]\(2, -2.2, 2.42, -2.662, 2.9282, \ldots\)[/tex]
- Compute differences between consecutive terms:
- [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 arithmetic.
3. Sequence: [tex]\(5, 1, -3, -7, -11, \ldots\)[/tex]
- Compute differences between consecutive terms:
- [tex]\( 1 - 5 = -4 \)[/tex]
- [tex]\( -3 - 1 = -4 \)[/tex]
- [tex]\( -7 - (-3) = -4 \)[/tex]
- [tex]\( -11 - (-7) = -4 \)[/tex]
- Since the differences are all the same (-4), this sequence is arithmetic.
4. Sequence: [tex]\(-3, 3, 9, 15, 21, \ldots\)[/tex]
- Compute differences between consecutive terms:
- [tex]\( 3 - (-3) = 6 \)[/tex]
- [tex]\( 9 - 3 = 6 \)[/tex]
- [tex]\( 15 - 9 = 6 \)[/tex]
- [tex]\( 21 - 15 = 6 \)[/tex]
- Since the differences are all the same (6), this sequence is arithmetic.
5. Sequence: [tex]\(-6.2, -3.1, -1.55, -0.775, -0.3875, \ldots\)[/tex]
- Compute differences between consecutive terms:
- [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 arithmetic.
Based on the above computations, the sequences that are arithmetic are:
- The third sequence: [tex]\(5, 1, -3, -7, -11, \ldots\)[/tex]
- The fourth sequence: [tex]\(-3, 3, 9, 15, 21, \ldots\)[/tex]
Therefore, the correct sequences that are arithmetic are:
[tex]\[ 3, 4 \][/tex]
Let's carefully analyze each of the given sequences to identify if they are arithmetic or not.
1. Sequence: [tex]\(-8.6, -5.0, -1.4, 2.2, 5.8, \ldots\)[/tex]
- Compute differences between consecutive terms:
- [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 the differences are all the same (3.6), this sequence is arithmetic.
2. Sequence: [tex]\(2, -2.2, 2.42, -2.662, 2.9282, \ldots\)[/tex]
- Compute differences between consecutive terms:
- [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 arithmetic.
3. Sequence: [tex]\(5, 1, -3, -7, -11, \ldots\)[/tex]
- Compute differences between consecutive terms:
- [tex]\( 1 - 5 = -4 \)[/tex]
- [tex]\( -3 - 1 = -4 \)[/tex]
- [tex]\( -7 - (-3) = -4 \)[/tex]
- [tex]\( -11 - (-7) = -4 \)[/tex]
- Since the differences are all the same (-4), this sequence is arithmetic.
4. Sequence: [tex]\(-3, 3, 9, 15, 21, \ldots\)[/tex]
- Compute differences between consecutive terms:
- [tex]\( 3 - (-3) = 6 \)[/tex]
- [tex]\( 9 - 3 = 6 \)[/tex]
- [tex]\( 15 - 9 = 6 \)[/tex]
- [tex]\( 21 - 15 = 6 \)[/tex]
- Since the differences are all the same (6), this sequence is arithmetic.
5. Sequence: [tex]\(-6.2, -3.1, -1.55, -0.775, -0.3875, \ldots\)[/tex]
- Compute differences between consecutive terms:
- [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 arithmetic.
Based on the above computations, the sequences that are arithmetic are:
- The third sequence: [tex]\(5, 1, -3, -7, -11, \ldots\)[/tex]
- The fourth sequence: [tex]\(-3, 3, 9, 15, 21, \ldots\)[/tex]
Therefore, the correct sequences that are arithmetic are:
[tex]\[ 3, 4 \][/tex]
