Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.