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

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

1 answer

To determine which polynomial sequence has constant second differences, we need to calculate the first differences and then the second differences for each of the given sequences.

Let's go through each sequence step by step.

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

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

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

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

  3. Sequence: {-6, -1, 14, 39}

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

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

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

From this analysis, we can conclude that the polynomial sequence that shows constant second differences is {-6, -1, 14, 39}.