Question

Which polynomial sequence shows that the terms of the 2nd differences are constant? (1 point) Responses {−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 {1,7,12,19} left brace 1 comma 7 comma 12 comma 19 right brace {2,6,11,14}

Answer 1

To determine which polynomial sequence shows constant second differences, we can calculate the first and second differences for each sequence.

1. For the sequence \(\{-6, -1, 14, 39\}\):
- First differences:
- \(-1 - (-6) = 5\)
- \(14 - (-1) = 15\)
- \(39 - 14 = 25\)
- First differences: \(\{5, 15, 25\}\)

- Second differences:
- \(15 - 5 = 10\)
- \(25 - 15 = 10\)
- Second differences: \(\{10, 10\}\) (constant)

2. For the sequence \(\{0, 4, 16, 34\}\):
- First differences:
- \(4 - 0 = 4\)
- \(16 - 4 = 12\)
- \(34 - 16 = 18\)
- First differences: \(\{4, 12, 18\}\)

- Second differences:
- \(12 - 4 = 8\)
- \(18 - 12 = 6\)
- Second differences: \(\{8, 6\}\) (not constant)

3. For the sequence \(\{1, 7, 12, 19\}\):
- First differences:
- \(7 - 1 = 6\)
- \(12 - 7 = 5\)
- \(19 - 12 = 7\)
- First differences: \(\{6, 5, 7\}\)

- Second differences:
- \(5 - 6 = -1\)
- \(7 - 5 = 2\)
- Second differences: \(\{-1, 2\}\) (not constant)

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

- Second differences:
- \(5 - 4 = 1\)
- \(3 - 5 = -2\)
- Second differences: \(\{1, -2\}\) (not constant)

The only sequence that has constant second differences is \(\{-6, -1, 14, 39\}\).