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

Question

Responses

{1,7,12,19}
left brace 1 comma 7 comma 12 comma 19 right brace

{2,6,11,14}
left brace 2 comma 6 comma 11 comma 14 right brace

{0,4,16,34}
left brace 0 comma 4 comma 16 comma 34 right brace

{−6,−1,14,39}

Answer 1

To determine which polynomial sequence has constant second differences, we will first calculate the first differences for each sequence and then find the second differences.

1. For the sequence {1, 7, 12, 19}:

   • First differences: 7 - 1 = 6, 12 - 7 = 5, 19 - 12 = 7 → {6, 5, 7}
   • Second differences: 5 - 6 = -1, 7 - 5 = 2 → {-1, 2}

2. For the sequence {2, 6, 11, 14}:

   • First differences: 6 - 2 = 4, 11 - 6 = 5, 14 - 11 = 3 → {4, 5, 3}
   • Second differences: 5 - 4 = 1, 3 - 5 = -2 → {1, -2}

3. For the sequence {0, 4, 16, 34}:

   • First differences: 4 - 0 = 4, 16 - 4 = 12, 34 - 16 = 18 → {4, 12, 18}
   • Second differences: 12 - 4 = 8, 18 - 12 = 6 → {8, 6}

4. For the sequence {−6, −1, 14, 39}:

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

From this analysis, the sequence {−6, −1, 14, 39} shows constant second differences (both are 10).

Thus, the correct answer is: \({-6, -1, 14, 39}\)