Which sequence has constant 2nd differences? (1 point)
Responses
{5, 11.5, 18.5, 24.5}
{5, 11.5, 18.5, 24.5}
{0.5, 5, 12.5, 23}
{0.5, 5, 12.5, 23}
{−0.5, 5, 13.5, 23}
{−0.5, 5, 13.5, 23}
{5, 13.5, 23, 34.5}
{5, 13.5, 23, 34.5}
To determine which sequence has constant 2nd differences, we need to calculate the differences between consecutive terms and then the differences between those differences.
Looking at the differences between consecutive terms for each sequence:
Sequence 1: 11.5 - 5 = 6.5, 18.5 - 11.5 = 7, 24.5 - 18.5 = 6
Sequence 2: 5 - 0.5 = 4.5, 12.5 - 5 = 7.5, 23 - 12.5 = 10.5
Sequence 3: 5 - (-0.5) = 5.5, 13.5 - 5 = 8.5, 23 - 13.5 = 9.5
Sequence 4: 13.5 - 5 = 8.5, 23 - 13.5 = 9.5, 34.5 - 23 = 11.5
Now let's calculate the differences between these differences:
Sequence 1: 7 - 6.5 = 0.5, 6 - 7 = -1
Sequence 2: 7.5 - 4.5 = 3, 10.5 - 7.5 = 3
Sequence 3: 8.5 - 5.5 = 3, 9.5 - 8.5 = 1
Sequence 4: 9.5 - 8.5 = 1, 11.5 - 9.5 = 2
Based on these calculations, we can see that Sequence 1, Sequence 3, and Sequence 4 have constant 2nd differences, as the differences between the differences are all the same. Therefore, the correct answer is:
{5, 18.5, 19.5}
To check for constant 2nd differences in a sequence, you need to find the differences between consecutive terms, and then the differences between those differences. Let's go through each of the sequences and check for constant 2nd differences.
1) {5, 11.5, 18.5, 24.5}
- Differences between consecutive terms: 11.5 - 5 = 6.5, 18.5 - 11.5 = 7, 24.5 - 18.5 = 6
- Differences between those differences: 7 - 6.5 = 0.5, 6 - 7 = -1
- Since the 2nd differences are not constant, this sequence does not have constant 2nd differences.
2) {0.5, 5, 12.5, 23}
- Differences between consecutive terms: 5 - 0.5 = 4.5, 12.5 - 5 = 7.5, 23 - 12.5 = 10.5
- Differences between those differences: 7.5 - 4.5 = 3, 10.5 - 7.5 = 3
- The 2nd differences are constant, so this sequence has constant 2nd differences.
3) {−0.5, 5, 13.5, 23}
- Differences between consecutive terms: 5 - (-0.5) = 5.5, 13.5 - 5 = 8.5, 23 - 13.5 = 9.5
- Differences between those differences: 8.5 - 5.5 = 3, 9.5 - 8.5 = 1
- The 2nd differences are not constant, so this sequence does not have constant 2nd differences.
4) {5, 13.5, 23, 34.5}
- Differences between consecutive terms: 13.5 - 5 = 8.5, 23 - 13.5 = 9.5, 34.5 - 23 = 11.5
- Differences between those differences: 9.5 - 8.5 = 1, 11.5 - 9.5 = 2
- The 2nd differences are not constant, so this sequence does not have constant 2nd differences.
Therefore, the sequence {0.5, 5, 12.5, 23} is the only sequence with constant 2nd differences out of the given options.