To construct an approximate polynomial expression for a sequence with constant 4th differences of 48, we can follow these steps:
Step 1: Understand the nature of the polynomial
Since the given sequence has constant 4th differences, this suggests that the sequence can be represented as a polynomial of degree 4 (quartic polynomial). A general polynomial of degree 4 is expressed as:
\[ P(n) = an^4 + bn^3 + cn^2 + dn + e \]
Step 2: Determine the value of \(a\)
The constant difference gives us information about the leading coefficient \(a\) of the polynomial. The \(k\)th differences of a polynomial are related to the coefficients of that polynomial such that:
- The 0th difference is simply the sequence itself.
- The 1st difference is the difference of consecutive terms.
- The 2nd difference is the difference of the 1st differences.
- The 3rd difference is the difference of the 2nd differences.
- The 4th difference is constant.
For a 4th-degree polynomial, the 4th difference is given by:
\[ 4! \cdot a = 24a \]
Given that the 4th difference is 48, we set up the equation:
\[ 24a = 48 \]
Solving for \(a\):
\[ a = \frac{48}{24} = 2 \]
Step 3: Use the known differences to find other coefficients
Next, we need the second term of the polynomial. To do this, we can compute the 3rd, 2nd, and 1st differences to find coefficients \(b\), \(c\), \(d\), and \(e\).
For a polynomial,
- The 3rd difference will be related to \(b\),
- The 2nd difference will involve \(c\),
- The 1st difference will involve \(d\).
Step 4: Calculate the coefficients from initial conditions
To find specific values for coefficients \(b\), \(c\), \(d\), and \(e\), you typically need initial values from the sequence. Let’s assume we have the following first few terms of the sequence:
- \(P(1) = A\)
- \(P(2) = B\)
- \(P(3) = C\)
- \(P(4) = D\)
We can set up a system of equations based on these values:
\[ \begin{align*} P(1) & = 2(1)^4 + b(1)^3 + c(1)^2 + d(1) + e = 2 + b + c + d + e = A \ P(2) & = 2(2)^4 + b(2)^3 + c(2)^2 + d(2) + e = 32 + 8b + 4c + 2d + e = B \ P(3) & = 2(3)^4 + b(3)^3 + c(3)^2 + d(3) + e = 162 + 27b + 9c + 3d + e = C \ P(4) & = 2(4)^4 + b(4)^3 + c(4)^2 + d(4) + e = 512 + 64b + 16c + 4d + e = D \ \end{align*} \]
This generates a system of equations that can be solved simultaneously to find the values of \(b\), \(c\), \(d\), and \(e\).
Step 5: Solve the system of equations
By substituting the values from our assumed sequence into our polynomial equations, we theoretically solve for \(b\), \(c\), \(d\), and \(e\).
Step 6: Formulate the polynomial expression
Once we have the coefficients \(a\), \(b\), \(c\), \(d\), and \(e\), we can write the full polynomial expression:
\[ P(n) = 2n^4 + bn^3 + cn^2 + dn + e \]
This results in an approximate polynomial expression representing the given sequence with constant 4th differences of 48. The specific values can vary depending on the provided terms of the sequence.
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