To construct a polynomial expression that approximates a sequence with a constant fourth difference, we first need to understand the implications of the constant difference and the degree of the polynomial we are working with.
Step 1: Understanding the Sequence
A sequence with a constant fourth difference implies that its generating polynomial must be a polynomial of degree 4. This is because the k-th difference of a polynomial of degree \( n \) becomes constant when taken \( k \) times, where \( k \) exceeds \( n \). Since we have a constant fourth difference, we know our polynomial must be of the form:
\[ P(n) = a_4 n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0 \]
Step 2: Establish the Fourth Difference
Given that the constant fourth difference of the sequence is 48, we can use the relationship between the coefficients of the polynomial and the constant differences.
The fourth difference of a polynomial of degree 4 is related to the leading coefficient \( a_4 \) by the formula: \[ \Delta^4 P(n) = 24a_4 \] where \( \Delta^4 \) denotes the fourth forward difference. To find \( a_4 \), we can set up the equation:
\[ 24a_4 = 48 \]
Solving for \( a_4 \):
\[ a_4 = \frac{48}{24} = 2 \]
Step 3: Determine the Remaining Coefficients
At this point, we have:
\[ P(n) = 2n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0 \]
To find the remaining coefficients \( a_3, a_2, a_1, \) and \( a_0 \), we typically require additional information from the sequence, such as specific values. Suppose we have the first few terms of the sequence (e.g., \( P(1), P(2), \) etc.), we can set up a system of equations using these terms.
For example, if we know:
- \( P(1) = x_1 \)
- \( P(2) = x_2 \)
- \( P(3) = x_3 \)
- \( P(4) = x_4 \)
We can plug in these values into the polynomial expression:
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For \( n = 1 \): \[ P(1) = 2(1)^4 + a_3(1)^3 + a_2(1)^2 + a_1(1) + a_0 = 2 + a_3 + a_2 + a_1 + a_0 = x_1 \]
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For \( n = 2 \): \[ P(2) = 2(2)^4 + a_3(2)^3 + a_2(2)^2 + a_1(2) + a_0 = 32 + 8a_3 + 4a_2 + 2a_1 + a_0 = x_2 \]
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For \( n = 3 \): \[ P(3) = 2(3)^4 + a_3(3)^3 + a_2(3)^2 + a_1(3) + a_0 = 162 + 27a_3 + 9a_2 + 3a_1 + a_0 = x_3 \]
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For \( n = 4 \): \[ P(4) = 2(4)^4 + a_3(4)^3 + a_2(4)^2 + a_1(4) + a_0 = 512 + 64a_3 + 16a_2 + 4a_1 + a_0 = x_4 \]
By solving this system of equations, you can find the values of \( a_3, a_2, a_1, \) and \( a_0 \).
Summary
- Establish that the polynomial is of degree 4 because of the constant fourth difference.
- Use the relationship \( 24a_4 = 48 \) to find \( a_4 = 2 \).
- Gather values from the sequence to set up equations for \( a_3, a_2, a_1, \) and \( a_0 \) by substituting terms of the sequence into the polynomial.
- Solve the system of equations to find all coefficients.
This will give you the polynomial expression that represents the sequence with a constant fourth difference of 48.