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

Answer 1

To construct a polynomial expression that represents a sequence with constant 4th differences, we can follow these steps:

Step 1: Identify the polynomial degree

A sequence has constant \(n\)th differences if it can be represented by a polynomial of degree \(n\). Since the 4th differences are constant in this case, the polynomial will be of degree 4.

Step 2: General form of the polynomial

The general form of a polynomial of degree 4 is:

\[ P(n) = an^4 + bn^3 + cn^2 + dn + e \]

where \(a\), \(b\), \(c\), \(d\), and \(e\) are coefficients we need to determine.

Step 3: Find the leading coefficient

The constant 4th differences imply that the 4th difference of the polynomial is constant, specifically, equal to \(48\). The formula for the constant \(k\) for a 4th-degree polynomial's leading term based on the constant 4th difference is:

\[ k = 24a \]

where \(k\) is the constant value of the 4th difference (which is given as 48).

Solving for \(a\):

\[ 24a = 48 \implies a = \frac{48}{24} = 2 \]

So we have:

\[ P(n) = 2n^4 + bn^3 + cn^2 + dn + e \]

Step 4: Express the first four differences

For any polynomial \(P(n)\), the \(k\)-th difference can be expressed in terms of the polynomial coefficients. To find the remaining coefficients \(b\), \(c\), \(d\), and \(e\), additional information (such as initial terms of the sequence) is needed.

Assuming we have values for the sequence \( P(1), P(2), P(3), \) and \( P(4) \), we can calculate those initial terms and then:

1. Construct a table of values for \(P(n)\).
2. Calculate the first differences by subtracting consecutive \(P(n)\) values.
3. Calculate the second differences from the first differences, and then the third and fourth differences.
4. Set the 4th difference to \(48\) and ensure it remains constant.

Step 5: Set initial conditions

Assuming we have specific values for \(P(1)\), \(P(2)\), \(P(3)\), and \(P(4)\), we can form a system of equations by selecting these values:

\[ \begin{align*} P(1) &= 2(1)^4 + b(1)^3 + c(1)^2 + d(1) + e \ P(2) &= 2(2)^4 + b(2)^3 + c(2)^2 + d(2) + e \ P(3) &= 2(3)^4 + b(3)^3 + c(3)^2 + d(3) + e \ P(4) &= 2(4)^4 + b(4)^3 + c(4)^2 + d(4) + e \end{align*} \]

This leads to a system of linear equations that you can solve to find \(b\), \(c\), \(d\), and \(e\).

Step 6: Solve the system of equations

Using a numerical or algebraic method, you can solve this system to determine the coefficients \(b\), \(c\), \(d\), and \(e\).

Conclusion

Once you determine all coefficients, the polynomial expression for the sequence can be written in its complete form:

\[ P(n) = 2n^4 + bn^3 + cn^2 + dn + e \]

This polynomial will generate a sequence with a constant 4th difference of \(48\).