To determine the degree of a polynomial sequence such that its 6th differences are constant, we need to understand the relationship between the degree of a polynomial and the constant nature of its finite differences.
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Understanding Finite Differences: The \( k \)-th finite difference of a sequence is obtained by taking the differences of the previous sequence \( k \) times.
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Polynomial Degree and Finite Differences: For a polynomial of degree \( n \):
- The \( (n+1) \)-th differences are constant.
- Therefore, if we want the 6th differences to be constant, we need to find \( n \) such that \( n + 1 = 6 \).
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Solving for \( n \): \[ n + 1 = 6 \implies n = 6 - 1 = 5. \]
Thus, the degree of the polynomial sequence has to be \( 5 \) if its 6th differences are to be a constant value.
In conclusion, a polynomial sequence must be of degree 5 for its 6th differences to be constant.
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