Question

which polynomial sequence shows that the terms of the 2nd differences are constant?
1,7,12,19
0,4,16,34
2,6,11,14
-6,-1,14,39

Answer 1

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.

1. **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**

2. **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**

3. **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**

4. **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**

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).