To determine which polynomial sequence shows that the terms of the second differences are constant, we first need to compute the first and second differences for each of the given sequences.
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First sequence: 1, 7, 12, 19
- First Differences:
- 7 - 1 = 6
- 12 - 7 = 5
- 19 - 12 = 7
- So, First differences: 6, 5, 7
- Second Differences:
- 5 - 6 = -1
- 7 - 5 = 2
- So, Second differences: -1, 2
- First Differences:
-
Second sequence: 0, 4, 16, 34
- First Differences:
- 4 - 0 = 4
- 16 - 4 = 12
- 34 - 16 = 18
- So, First differences: 4, 12, 18
- Second Differences:
- 12 - 4 = 8
- 18 - 12 = 6
- So, Second differences: 8, 6
- First Differences:
-
Third sequence: 2, 6, 11, 14
- First Differences:
- 6 - 2 = 4
- 11 - 6 = 5
- 14 - 11 = 3
- So, First differences: 4, 5, 3
- Second Differences:
- 5 - 4 = 1
- 3 - 5 = -2
- So, Second differences: 1, -2
- First Differences:
-
Fourth sequence: -6, -1, 14, 39
- First Differences:
- -1 - (-6) = 5
- 14 - (-1) = 15
- 39 - 14 = 25
- So, First differences: 5, 15, 25
- Second Differences:
- 15 - 5 = 10
- 25 - 15 = 10
- So, Second differences: 10, 10
- First Differences:
Now, we observe the second differences for each sequence:
- First Sequence: -1, 2 (not constant)
- Second Sequence: 8, 6 (not constant)
- Third Sequence: 1, -2 (not constant)
- Fourth Sequence: 10, 10 (constant)
The fourth sequence (-6, -1, 14, 39) is the only one that has constant second differences. Therefore, it shows that the terms have a polynomial progression of degree 2 (quadratic).