Question

Which sequence has constant 2nd differences?
{5, 11.5, 18.5, 24.5}
{-0.5, 5, 13.5, 23}
{5, 13.5, 23, 34.5}
{0.5, 5, 12.5, 23}

Answer 1

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

**1. 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)

**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: {5.5, 8.5, 9.5}

- Second Differences:
- 8.5 - 5.5 = 3
- 9.5 - 8.5 = 1
- Second differences: {3, 1} (not constant)

**3. 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)

**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)

Based on this analysis, the sequence that has constant second differences is **{0.5, 5, 12.5, 23}**.