To construct a polynomial expression that represents a sequence with constant 4th differences of 48, we follow these steps:
Step 1: Understand the Nature of the Polynomial
Given that the sequence has constant 4th differences, we know that the corresponding polynomial is of degree 4. We can express this polynomial in the general form:
\[ P(n) = a_4 n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0 \]
where \( a_4, a_3, a_2, a_1, \) and \( a_0 \) are coefficients that we will determine.
Step 2: Determine the Leading Coefficient
Since the 4th difference is constant and equal to 48, we can determine the leading coefficient \( a_4 \).
The k-th finite difference of a polynomial \( P(n) \) of degree \( k \) is constant and equal to \( k! \times a_k \). For a polynomial of degree 4, the 4th difference is given by:
\[ \Delta^4 P(n) = 4! \cdot a_4 = 24a_4 \]
Given that \( \Delta^4 P(n) = 48 \), we can set up the equation:
\[ 24a_4 = 48 \]
Solving for \( a_4 \):
\[ a_4 = \frac{48}{24} = 2 \]
Step 3: Polynomial Expression Up to a Point
Now we know the leading coefficient, so we can express our polynomial as:
\[ P(n) = 2n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0 \]
Step 4: Determine Additional Coefficients
To find the remaining coefficients \( a_3, a_2, a_1, a_0 \), we need more information about the sequence. Typically, this information can be provided in the form of initial values of the sequence at certain points, such as:
- \( P(0) = s_0 \)
- \( P(1) = s_1 \)
- \( P(2) = s_2 \)
- \( P(3) = s_3 \)
Where \( s_0, s_1, s_2, s_3 \) are known values of the sequence at points 0, 1, 2, and 3.
Step 5: Set Up a System of Equations
Using the known values of the sequence, we can set up equations. For instance, using generic values,
- \( P(0) = a_0 = s_0 \)
- \( P(1) = 2(1)^4 + a_3(1)^3 + a_2(1)^2 + a_1(1) + a_0 = s_1 \)
- \( P(2) = 2(2)^4 + a_3(2)^3 + a_2(2)^2 + a_1(2) + a_0 = s_2 \)
- \( P(3) = 2(3)^4 + a_3(3)^3 + a_2(3)^2 + a_1(3) + a_0 = s_3 \)
You will end up with a system of four equations with four unknowns \( a_3, a_2, a_1, a_0 \).
Step 6: Solve the System of Equations
By substituting known values into the equations, you can solve the system to find the values of the remaining coefficients.
Final Step: Formulate the Polynomial
Substituting the values of \( a_3, a_2, a_1, \) and \( a_0 \) back into the polynomial expression gives you the desired 4th-degree polynomial that approximates the sequence with constant 4th differences of 48.
Example
For instance, if you knew \( P(0) = 1, P(1) = 3, P(2) = 9, P(3) = 19 \), you could substitute these values into the equations to find \( a_3, a_2, a_1, a_0 \) accordingly.
By following these steps, you can construct a polynomial expression that effectively represents the concerned sequence.