Let's analyze the given sequences step-by-step.
The first row of the table represents the first sequence:
[tex]\[0, 3, 6, 9, 12\][/tex]
The second row of the table represents the second sequence:
[tex]\[-1, 5, 11, 17, 23\][/tex]
We now want to understand how each element in the first sequence maps to an element in the second sequence. We can consider each column and look for any relation or pattern.
1. The first pair of numbers is (0, -1):
- 0 corresponds to -1.
2. The second pair of numbers is (3, 5):
- 3 corresponds to 5.
3. The third pair of numbers is (6, 11):
- 6 corresponds to 11.
4. The fourth pair of numbers is (9, 17):
- 9 corresponds to 17.
5. The fifth pair of numbers is (12, 23):
- 12 corresponds to 23.
To understand the pattern, we can observe that each element in the second sequence depends on the corresponding element in the first sequence. Let's see if there is a consistent pattern by finding the difference between the corresponding terms in the sequences:
1. [tex]\(-1 - 0 = -1\)[/tex]
2. [tex]\(5 - 3 = 2\)[/tex]
3. [tex]\(11 - 6 = 5\)[/tex]
4. [tex]\(17 - 9 = 8\)[/tex]
5. [tex]\(23 - 12 = 11\)[/tex]
By examining these differences, we notice that they increase by 3 each time:
- The first difference [tex]\((-1 - 0)\)[/tex] is [tex]\(-1\)[/tex].
- The second difference [tex]\((5 - 3)\)[/tex] is 2, which is 3 more than [tex]\(-1\)[/tex].
- The third difference [tex]\((11 - 6)\)[/tex] is 5, which is 3 more than 2.
- The fourth difference [tex]\((17 - 9)\)[/tex] is 8, which is 3 more than 5.
- The fifth difference [tex]\((23 - 12)\)[/tex] is 11, which is 3 more than 8.
This suggests that the second sequence can be generated from the first sequence by applying a set rule. Specifically, it appears that the difference from one element in the second sequence to the corresponding element in the first increases by 3 with each successive term.
Therefore, the sequences are established as follows:
- The first sequence is [tex]\(0, 3, 6, 9, 12\)[/tex].
- The second sequence is [tex]\(-1, 5, 11, 17, 23\)[/tex].
Each element of the second sequence is generated by adding an incrementally increasing value to the corresponding element of the first sequence, starting with [tex]\(-1\)[/tex] for the first element.
