Question

Which polynomial sequence shows that the terms of the 2nd differences are constant? (1 point)
Responses

{1,7,12,19}
left brace 1 comma 7 comma 12 comma 19 right brace

{−6,−1,14,39}
left brace negative 6 comma negative 1 comma 14 comma 39 right brace

{0,4,16,34}
left brace 0 comma 4 comma 16 comma 34 right brace

{2,6,11,14}

Answer 1

To determine which polynomial sequence has constant second differences, we need to calculate the first differences and then the second differences for each sequence.

**1. Sequence: {1, 7, 12, 19}**
- First differences: \(7 - 1 = 6\), \(12 - 7 = 5\), \(19 - 12 = 7\) → {6, 5, 7}
- Second differences: \(5 - 6 = -1\), \(7 - 5 = 2\) → {-1, 2} (not constant)

**2. Sequence: {-6, -1, 14, 39}**
- First differences: \(-1 - (-6) = 5\), \(14 - (-1) = 15\), \(39 - 14 = 25\) → {5, 15, 25}
- Second differences: \(15 - 5 = 10\), \(25 - 15 = 10\) → {10, 10} (constant)

**3. Sequence: {0, 4, 16, 34}**
- First differences: \(4 - 0 = 4\), \(16 - 4 = 12\), \(34 - 16 = 18\) → {4, 12, 18}
- Second differences: \(12 - 4 = 8\), \(18 - 12 = 6\) → {8, 6} (not constant)

**4. Sequence: {2, 6, 11, 14}**
- First differences: \(6 - 2 = 4\), \(11 - 6 = 5\), \(14 - 11 = 3\) → {4, 5, 3}
- Second differences: \(5 - 4 = 1\), \(3 - 5 = -2\) → {1, -2} (not constant)

Based on the calculations, the polynomial sequence that shows constant second differences is:

**{-6, -1, 14, 39}**