Question

Explain the steps to construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48. make it short

Answer 1

To construct an approximate polynomial expression for a sequence with constant 4th differences of 48, follow these steps:

1. **Identify the Degree**: Since the 4th differences are constant, the polynomial will be of degree 4.

2. **General Form**: Start with the general form of a 4th degree polynomial:
\[
P(n) = an^4 + bn^3 + cn^2 + dn + e
\]

3. **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:
\[
24a = 48 \implies a = 2
\]

4. **Form the Polynomial**: Substitute \(a\) into the polynomial:
\[
P(n) = 2n^4 + bn^3 + cn^2 + dn + e
\]

5. **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.

6. **Final Polynomial**: After calculating the coefficients, you can present your final polynomial expression.

This approximates the sequence based on the constant 4th differences.