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

Responses

{5,11.5,18.5,24.5}
left brace 5 comma 11.5 comma 18.5 comma 24.5 right brace

{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

{5,13.5,23,34.5}

1 answer

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

  1. For the 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. For the 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)

  3. For the 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)

  4. For the 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)

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