Which sequence has constant 2nd differences? (1 point)

Question

Responses

{5, 11.5, 18.5, 24.5}
{5, 11.5, 18.5, 24.5}

{5, 13.5, 23, 34.5}
{5, 13.5, 23, 34.5}

{0.5, 5, 12.5, 23}
{0.5, 5, 12.5, 23}

{-0.5, 5, 13.5, 23}
{-0.5, 5, 13.5, 23}

Answer 1

To determine which sequence has constant second differences, we first need to calculate the first and 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: {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)

3. 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)

4. 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)

Based on the calculations:

• Only the sequence {0.5, 5, 12.5, 23} has constant second differences of 3.

So, the correct response is:

{0.5, 5, 12.5, 23}