Answer :
Sure, let's determine the next three terms for each given sequence:
### Sequence 1: [tex]\(2, 5, 8, 11, \ldots\)[/tex]
This is an arithmetic sequence with a common difference. To find the common difference ([tex]\(d\)[/tex]):
[tex]\[d = 5 - 2 = 3\][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= 11 + 3 = 14 \\ \text{6th term} &= 14 + 3 = 17 \\ \text{7th term} &= 17 + 3 = 20 \end{align*} \][/tex]
### Next three terms: [tex]\(14, 17, 20\)[/tex]
### Sequence 2: [tex]\(1, -3, -7, -11, \ldots\)[/tex]
This is an arithmetic sequence with a common difference. To find the common difference ([tex]\(d\)[/tex]):
[tex]\[d = -3 - 1 = -4\][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= -11 - 4 = -15 \\ \text{6th term} &= -15 - 4 = -19 \\ \text{7th term} &= -19 - 4 = -23 \end{align*} \][/tex]
### Next three terms: [tex]\(-15, -19, -23\)[/tex]
### Sequence 3: [tex]\(1, 4, 16, 64, \ldots\)[/tex]
This is a geometric sequence with a common ratio. To find the common ratio ([tex]\(r\)[/tex]):
[tex]\[ r = \frac{4}{1} = 4 \][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= 64 \times 4 = 256 \\ \text{6th term} &= 256 \times 4 = 1024 \\ \text{7th term} &= 1024 \times 4 = 4096 \end{align*} \][/tex]
### Next three terms: [tex]\(256, 1024, 4096\)[/tex]
### Sequence 4: [tex]\(60, 48, 36, 24, \ldots\)[/tex]
This is an arithmetic sequence with a common difference. To find the common difference ([tex]\(d\)[/tex]):
[tex]\[d = 48 - 60 = -12\][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= 24 - 12 = 12 \\ \text{6th term} &= 12 - 12 = 0 \\ \text{7th term} &= 0 - 12 = -12 \end{align*} \][/tex]
### Next three terms: [tex]\(12, 0, -12\)[/tex]
### Sequence 5: [tex]\(1, 4, 9, 16, 25, \ldots\)[/tex]
This is a quadratic sequence where each term is [tex]\(n^2\)[/tex].
The next three terms:
[tex]\[ \begin{align*} \text{6th term} &= 6^2 = 36 \\ \text{7th term} &= 7^2 = 49 \\ \text{8th term} &= 8^2 = 64 \end{align*} \][/tex]
### Next three terms: [tex]\(36, 49, 64\)[/tex]
### Sequence 6: [tex]\(2, 4, 8, 16, \ldots\)[/tex]
This is a geometric sequence with a common ratio. To find the common ratio ([tex]\(r\)[/tex]):
[tex]\[ r = \frac{4}{2} = 2 \][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= 16 \times 2 = 32 \\ \text{6th term} &= 32 \times 2 = 64 \\ \text{7th term} &= 64 \times 2 = 128 \end{align*} \][/tex]
### Next three terms: [tex]\(32, 64, 128\)[/tex]
### Sequence 7: [tex]\(2, 6, 18, 54, \ldots\)[/tex]
This is a geometric sequence with a common ratio. To find the common ratio ([tex]\(r\)[/tex]):
[tex]\[ r = \frac{6}{2} = 3 \][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= 54 \times 3 = 162 \\ \text{6th term} &= 162 \times 3 = 486 \\ \text{7th term} &= 486 \times 3 = 1458 \end{align*} \][/tex]
### Next three terms: [tex]\(162, 486, 1458\)[/tex]
### Sequence 8: [tex]\(3.2, 4.3, 5.4, \ldots\)[/tex]
This is an arithmetic sequence with a common difference. To find the common difference ([tex]\(d\)[/tex]):
[tex]\[d = 4.3 - 3.2 = 1.1\][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{4th term} &= 5.4 + 1.1 = 6.5 \\ \text{5th term} &= 6.5 + 1.1 = 7.6 \\ \text{6th term} &= 7.6 + 1.1 = 8.7 \end{align*} \][/tex]
### Next three terms: [tex]\(6.5, 7.6, 8.7\)[/tex]
### Sequence 9: [tex]\(-7, -9, -11, -13, \ldots\)[/tex]
This is an arithmetic sequence with a common difference. To find the common difference ([tex]\(d\)[/tex]):
[tex]\[d = -9 - (-7) = -2\][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= -13 - 2 = -15 \\ \text{6th term} &= -15 - 2 = -17 \\ \text{7th term} &= -17 - 2 = -19 \end{align*} \][/tex]
### Next three terms: [tex]\(-15, -17, -19\)[/tex]
That's the step-by-step solution for finding the next three terms of the given sequences!
### Sequence 1: [tex]\(2, 5, 8, 11, \ldots\)[/tex]
This is an arithmetic sequence with a common difference. To find the common difference ([tex]\(d\)[/tex]):
[tex]\[d = 5 - 2 = 3\][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= 11 + 3 = 14 \\ \text{6th term} &= 14 + 3 = 17 \\ \text{7th term} &= 17 + 3 = 20 \end{align*} \][/tex]
### Next three terms: [tex]\(14, 17, 20\)[/tex]
### Sequence 2: [tex]\(1, -3, -7, -11, \ldots\)[/tex]
This is an arithmetic sequence with a common difference. To find the common difference ([tex]\(d\)[/tex]):
[tex]\[d = -3 - 1 = -4\][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= -11 - 4 = -15 \\ \text{6th term} &= -15 - 4 = -19 \\ \text{7th term} &= -19 - 4 = -23 \end{align*} \][/tex]
### Next three terms: [tex]\(-15, -19, -23\)[/tex]
### Sequence 3: [tex]\(1, 4, 16, 64, \ldots\)[/tex]
This is a geometric sequence with a common ratio. To find the common ratio ([tex]\(r\)[/tex]):
[tex]\[ r = \frac{4}{1} = 4 \][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= 64 \times 4 = 256 \\ \text{6th term} &= 256 \times 4 = 1024 \\ \text{7th term} &= 1024 \times 4 = 4096 \end{align*} \][/tex]
### Next three terms: [tex]\(256, 1024, 4096\)[/tex]
### Sequence 4: [tex]\(60, 48, 36, 24, \ldots\)[/tex]
This is an arithmetic sequence with a common difference. To find the common difference ([tex]\(d\)[/tex]):
[tex]\[d = 48 - 60 = -12\][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= 24 - 12 = 12 \\ \text{6th term} &= 12 - 12 = 0 \\ \text{7th term} &= 0 - 12 = -12 \end{align*} \][/tex]
### Next three terms: [tex]\(12, 0, -12\)[/tex]
### Sequence 5: [tex]\(1, 4, 9, 16, 25, \ldots\)[/tex]
This is a quadratic sequence where each term is [tex]\(n^2\)[/tex].
The next three terms:
[tex]\[ \begin{align*} \text{6th term} &= 6^2 = 36 \\ \text{7th term} &= 7^2 = 49 \\ \text{8th term} &= 8^2 = 64 \end{align*} \][/tex]
### Next three terms: [tex]\(36, 49, 64\)[/tex]
### Sequence 6: [tex]\(2, 4, 8, 16, \ldots\)[/tex]
This is a geometric sequence with a common ratio. To find the common ratio ([tex]\(r\)[/tex]):
[tex]\[ r = \frac{4}{2} = 2 \][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= 16 \times 2 = 32 \\ \text{6th term} &= 32 \times 2 = 64 \\ \text{7th term} &= 64 \times 2 = 128 \end{align*} \][/tex]
### Next three terms: [tex]\(32, 64, 128\)[/tex]
### Sequence 7: [tex]\(2, 6, 18, 54, \ldots\)[/tex]
This is a geometric sequence with a common ratio. To find the common ratio ([tex]\(r\)[/tex]):
[tex]\[ r = \frac{6}{2} = 3 \][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= 54 \times 3 = 162 \\ \text{6th term} &= 162 \times 3 = 486 \\ \text{7th term} &= 486 \times 3 = 1458 \end{align*} \][/tex]
### Next three terms: [tex]\(162, 486, 1458\)[/tex]
### Sequence 8: [tex]\(3.2, 4.3, 5.4, \ldots\)[/tex]
This is an arithmetic sequence with a common difference. To find the common difference ([tex]\(d\)[/tex]):
[tex]\[d = 4.3 - 3.2 = 1.1\][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{4th term} &= 5.4 + 1.1 = 6.5 \\ \text{5th term} &= 6.5 + 1.1 = 7.6 \\ \text{6th term} &= 7.6 + 1.1 = 8.7 \end{align*} \][/tex]
### Next three terms: [tex]\(6.5, 7.6, 8.7\)[/tex]
### Sequence 9: [tex]\(-7, -9, -11, -13, \ldots\)[/tex]
This is an arithmetic sequence with a common difference. To find the common difference ([tex]\(d\)[/tex]):
[tex]\[d = -9 - (-7) = -2\][/tex]
The next three terms:
[tex]\[ \begin{align*} \text{5th term} &= -13 - 2 = -15 \\ \text{6th term} &= -15 - 2 = -17 \\ \text{7th term} &= -17 - 2 = -19 \end{align*} \][/tex]
### Next three terms: [tex]\(-15, -17, -19\)[/tex]
That's the step-by-step solution for finding the next three terms of the given sequences!
