To understand the concept of second differences and how they relate to polynomial sequences, let's first clarify what first and second differences are:
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First Differences: Given a sequence of terms \( a_0, a_1, a_2, \ldots \), the first differences are obtained by subtracting consecutive terms: \[ \Delta a_n = a_{n+1} - a_n \]
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Second Differences: The second differences are the differences of the first differences: \[ \Delta^2 a_n = \Delta a_{n+1} - \Delta a_n \]
Now, for polynomial sequences:
- Linear polynomials (degree 1) have constant first differences, which implies that their second differences are zero.
- Quadratic polynomials (degree 2) have constant second differences.
- Cubic polynomials (degree 3) have constant third differences.
The term "cibsrabt" seems to be a typographical error. If you meant "constant," then it is important to note that for a polynomial of degree 2, the second differences are constant.
Example of a Quadratic Polynomial
Consider the quadratic polynomial: \[ a_n = n^2 \]
Calculating the first differences: \[ \Delta a_n = (n+1)^2 - n^2 = 2n + 1 \]
Now, calculating the second differences: \[ \Delta^2 a_n = (2(n+1) + 1) - (2n + 1) = 2 \] The second differences are constant and equal to 2.
Conclusion
If you are looking for a polynomial sequence where the second differences are constant, then quadratic polynomials (such as \( a_n = n^2 \)) fit this criterion. If you have a different polynomial degree in mind or a specific sequence context, please provide more detail, and I can assist you further!