Question

Which Polynomial Sequence Shows That The Terms Of The 2nd Differences Are Constant? -6,-1,14,39
2,6,11,14
0,4,16,34
1,7,12,19

Answer 1

To determine which polynomial sequence shows that the terms of the 2nd differences are constant, we can examine the sequences provided.

1. **Calculate the first differences** for each sequence:
- For the sequence **-6, -1, 14, 39**:
- First differences: \(-1 - (-6) = 5\), \(14 - (-1) = 15\), \(39 - 14 = 25\)
- First differences: \(5, 15, 25\)

- For the sequence **2, 6, 11, 14**:
- First differences: \(6 - 2 = 4\), \(11 - 6 = 5\), \(14 - 11 = 3\)
- First differences: \(4, 5, 3\)

- For the sequence **0, 4, 16, 34**:
- First differences: \(4 - 0 = 4\), \(16 - 4 = 12\), \(34 - 16 = 18\)
- First differences: \(4, 12, 18\)

- For the sequence **1, 7, 12, 19**:
- First differences: \(7 - 1 = 6\), \(12 - 7 = 5\), \(19 - 12 = 7\)
- First differences: \(6, 5, 7\)

2. **Calculate the second differences** (the differences of the first differences):
- For the first sequence **5, 15, 25**:
- Second differences: \(15 - 5 = 10\), \(25 - 15 = 10\)
- Second differences: \(10, 10\) (constant)

- For the second sequence **4, 5, 3**:
- Second differences: \(5 - 4 = 1\), \(3 - 5 = -2\)
- Second differences: \(1, -2\) (not constant)

- For the third sequence **4, 12, 18**:
- Second differences: \(12 - 4 = 8\), \(18 - 12 = 6\)
- Second differences: \(8, 6\) (not constant)

- For the fourth sequence **6, 5, 7**:
- Second differences: \(5 - 6 = -1\), \(7 - 5 = 2\)
- Second differences: \(-1, 2\) (not constant)

From this analysis, only the first sequence (**-6, -1, 14, 39**) has constant second differences. Thus, this sequence is represented by a polynomial of degree 2.

So, the answer is: **-6, -1, 14, 39**.