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Which polynomial sequence shows that the terms of the 2nd differences are constant?

A. {1,7,12,19}
B. {−6,−1,14,39}
C. {0,4,16,34}
D. {0,4,16,34}

1 answer

To determine which polynomial sequence shows that the terms of the 2nd differences are constant, we first need to calculate the first differences and then the second differences for each sequence.

  1. Sequence A: {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)

  2. Sequence B: {−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)

  3. Sequence C: {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)

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

    • This is the same sequence as C, so we already know the result: not constant.

Based on the calculations, the only sequence where the 2nd differences are constant is:

B. {−6, −1, 14, 39}