Question

Which polynomial sequence shows that the terms of the 2nd differences are constant? (1 point)
{2,6,11,14}
{1,7,12,19}
{−6,−1,14,39}
{0,4,16,34}

Answer 1

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

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

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

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

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

The only sequence with constant second differences is:

**{−6, −1, 14, 39}**.