To determine at which differences the polynomial sequence \( a_n = 2n^4 - n^3 \) reaches a constant value, we can analyze the differences step by step.
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Determine the degree of the polynomial: The polynomial \( a_n = 2n^4 - n^3 \) is a degree 4 polynomial. Generally, for a polynomial of degree \( k \), the \( k \)th differences will reach a constant value.
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Calculate the differences:
- 1st differences: \( a_{n+1} - a_n \)
- 2nd differences: \( (a_{n+1} - a_n) - (a_n - a_{n-1}) \)
- 3rd differences: \( (2nd \ differences) - (2nd \ differences \ for \ n-1) \)
- 4th differences: \( (3rd \ differences) - (3rd \ differences \ for \ n-1) \)
Since we know from polynomial behavior that:
- The 1st differences of a polynomial of degree 4 will not be constant.
- The 2nd differences will not be constant.
- The 3rd differences will not be constant.
- The 4th differences will be constant.
Thus, for the polynomial \( a_n = 2n^4 - n^3 \), the sequence reaches a constant value at the 4th differences.
Final Answer: 4th differences
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