Final answer:
To find the value of k in the sequence, differences between terms were analyzed. The pattern of increasing differences by a factor of 42 was discovered, leading to the conclusion that the value of k is 26.
Step-by-step explanation:
The student is tasked with finding the value of k in the sequence 106, 102, 56, 20, k. This is a mathematics problem that requires identifying the pattern of the sequence to determine the next number. To solve this problem, let's first observe the differences between consecutive terms:
- 102 - 106 = -4
- 56 - 102 = -46
- 20 - 56 = -36
Neither the differences are constant (which would indicate an arithmetic sequence) nor do they form a recognizable pattern (such as a geometric sequence).
Let's consider the possibility of a more complex relationship and observe the differences of the differences:
- -46 - (-4) = -42
- -36 - (-46) = 10
This reveals that the change from one difference to the next is adding 42. If this pattern continues, we can add 42 to the last difference of -36 to get the next difference.
The next difference would thus be -36 + 42 = 6. Now, to find the next term k, we add this difference to the last known term:
Therefore, the value of k in the sequence is 26.