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Which polynomial sequence shows that the terms of the 2nd differences are constant? (1 point)

{2,6,11,14}
{1,7,12,19}
{−6,−1,14,39}
{0,4,16,34}

1 answer

To identify which polynomial sequence has constant second differences, we need to calculate the first and second differences for each sequence.

  1. Sequence {2, 6, 11, 14}:

    • First differences:
      • \(6 - 2 = 4\)
      • \(11 - 6 = 5\)
      • \(14 - 11 = 3\)
    • First difference sequence: {4, 5, 3}
    • Second differences:
      • \(5 - 4 = 1\)
      • \(3 - 5 = -2\)
    • Second difference sequence: {1, -2} (not constant)

  2. Sequence {1, 7, 12, 19}:

    • First differences:
      • \(7 - 1 = 6\)
      • \(12 - 7 = 5\)
      • \(19 - 12 = 7\)
    • First difference sequence: {6, 5, 7}
    • Second differences:
      • \(5 - 6 = -1\)
      • \(7 - 5 = 2\)
    • Second difference sequence: {-1, 2} (not constant)

  3. Sequence {−6, −1, 14, 39}:

    • First differences:
      • \(-1 - (-6) = 5\)
      • \(14 - (-1) = 15\)
      • \(39 - 14 = 25\)
    • First difference sequence: {5, 15, 25}
    • Second differences:
      • \(15 - 5 = 10\)
      • \(25 - 15 = 10\)
    • Second difference sequence: {10, 10} (constant)

  4. Sequence {0, 4, 16, 34}:

    • First differences:
      • \(4 - 0 = 4\)
      • \(16 - 4 = 12\)
      • \(34 - 16 = 18\)
    • First difference sequence: {4, 12, 18}
    • Second differences:
      • \(12 - 4 = 8\)
      • \(18 - 12 = 6\)
    • Second difference sequence: {8, 6} (not constant)

The only sequence with constant second differences is:

{−6, −1, 14, 39}.