Question

Which sequence has constant 2nd differences? (1 point)Responses{0.5,5,12.5,23}left brace 0.5 comma 5 comma 12.5 comma 23 right brace{−0.5,5,13.5,23}left brace negative 0.5 comma 5 comma 13.5 comma 23 right brace{5,11.5,18.5,24.5}left brace 5 comma 11.5 comma 18.5 comma 24.5 right brace{5,13.5,23,34.5}

Answer 1

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

1. **Sequence {0.5, 5, 12.5, 23}**
- First differences:
- 5 - 0.5 = 4.5
- 12.5 - 5 = 7.5
- 23 - 12.5 = 10.5
- First differences are {4.5, 7.5, 10.5}
- Second differences:
- 7.5 - 4.5 = 3
- 10.5 - 7.5 = 3
- Second differences are {3, 3}
- Constant second differences.

2. **Sequence {-0.5, 5, 13.5, 23}**
- First differences:
- 5 - (-0.5) = 5.5
- 13.5 - 5 = 8.5
- 23 - 13.5 = 9.5
- First differences are {5.5, 8.5, 9.5}
- Second differences:
- 8.5 - 5.5 = 3
- 9.5 - 8.5 = 1
- Second differences are {3, 1}
- Not constant.

3. **Sequence {5, 11.5, 18.5, 24.5}**
- First differences:
- 11.5 - 5 = 6.5
- 18.5 - 11.5 = 7
- 24.5 - 18.5 = 6
- First differences are {6.5, 7, 6}
- Second differences:
- 7 - 6.5 = 0.5
- 6 - 7 = -1
- Second differences are {0.5, -1}
- Not constant.

4. **Sequence {5, 13.5, 23, 34.5}**
- First differences:
- 13.5 - 5 = 8.5
- 23 - 13.5 = 9.5
- 34.5 - 23 = 11.5
- First differences are {8.5, 9.5, 11.5}
- Second differences:
- 9.5 - 8.5 = 1
- 11.5 - 9.5 = 2
- Second differences are {1, 2}
- Not constant.

The only sequence with constant second differences is the first sequence: **{0.5, 5, 12.5, 23}**.