To construct an approximate polynomial expression for a sequence with constant 4th differences of 48, follow these steps:
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Identify the Degree: Since the 4th differences are constant, the polynomial will be of degree 4.
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General Form: Start with the general form of a 4th degree polynomial: \[ P(n) = an^4 + bn^3 + cn^2 + dn + e \]
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Constant 4th Differences: The constant 4th differences of a polynomial \(P(n)\) can be related to the leading term: \[ 4!a = \text{constant 4th difference} \] Given that the 4th differences are 48, we set: \[ 24a = 48 \implies a = 2 \]
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Form the Polynomial: Substitute \(a\) into the polynomial: \[ P(n) = 2n^4 + bn^3 + cn^2 + dn + e \]
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Determine Coefficients: The coefficients \(b\), \(c\), \(d\), and \(e\) can be determined if you have specific values from the original sequence, by substituting known terms and solving the resulting system of equations.
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Final Polynomial: After calculating the coefficients, you can present your final polynomial expression.
This approximates the sequence based on the constant 4th differences.
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