Final answer:
The first sequence has no consistent first or second-level difference, indicating it is neither an arithmetic nor a quadratic sequence. The second sequence has a consistent first-level difference that increases by 1, suggesting a possible quadratic pattern.
Step-by-step explanation:
To identify the level of difference in a sequence, we calculate the differences between consecutive terms until we find a consistent pattern. Let's analyze each sequence provided:
Sequence a) 0, 12, 10, 0, -12, -20
Differences between terms:
- 12 - 0 = 12
- 10 - 12 = -2
- 0 - 10 = -10
- -12 - 0 = -12
-20 - (-12) = -8
Since there is no consistent pattern in the first set of differences, let's calculate the second set of differences to check for a second-level pattern:
- -2 - 12 = -14
- -10 - (-2) = -8
- -12 - (-10) = -2
- -8 - (-12) = 4
The second set of differences also does not show a pattern, indicating this sequence does not have a common difference (i.e., it is not an arithmetic sequence), nor does it show a consistent second-level difference (i.e., it is not a quadratic sequence).
Sequence b) 1, 4, 8, 13, 19, 26
Differences between terms:
- 4 - 1 = 3
- 8 - 4 = 4
- 13 - 8 = 5
- 19 - 13 = 6
- 26 - 19 = 7
The differences here are consistent and increase by 1 each time, indicating a
first-level difference. This pattern suggests the presence of a polynomial of degree higher than 1, possibly a quadratic pattern.