Question

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

Answer 1

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

The first difference is calculated by subtracting consecutive terms, and the second difference is calculated by subtracting consecutive first differences.

Let's compute this for each sequence:

1. **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: {8.5, 9.5, 11.5}
- Second differences:
- 9.5 - 8.5 = 1
- 11.5 - 9.5 = 2
- Second differences: {1, 2} → Not constant.

2. **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: {6.5, 7, 6}
- Second differences:
- 7 - 6.5 = 0.5
- 6 - 7 = -1
- Second differences: {0.5, -1} → Not constant.

3. **Sequence: {−0.5, 5, 12.5, 23}**
- First differences:
- 5 - (-0.5) = 5.5
- 12.5 - 5 = 7.5
- 23 - 12.5 = 10.5
- First differences: {5.5, 7.5, 10.5}
- Second differences:
- 7.5 - 5.5 = 2
- 10.5 - 7.5 = 3
- Second differences: {2, 3} → Not constant.

4. **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: {4.5, 7.5, 10.5}
- Second differences:
- 7.5 - 4.5 = 3
- 10.5 - 7.5 = 3
- Second differences: {3, 3} → Constant.

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