To create a polynomial expression for a sequence with constant 4th differences of 48, you can follow these steps:
Step 1: Understand the Concept of Differences
First, let's clarify what "constant 4th differences" means. When you look at a sequence of numbers, you can find the "first differences" by subtracting each number from the one that follows. If you do this repeatedly (getting second differences, third differences, and finally fourth differences), and if the fourth differences are constant, it indicates that the sequence can be represented by a polynomial of degree 4.
Step 2: Determine the Degree of the Polynomial
Since we have constant 4th differences, we know that the polynomial we are looking for is of degree 4. This means it can be written in the general form: \[ P(n) = an^4 + bn^3 + cn^2 + dn + e \] where \( a, b, c, d, \) and \( e \) are constants that we need to find.
Step 3: Use the Information about the 4th Differences
We know that the constant 4th difference is 48. This leads us to find the coefficient \( a \) in our polynomial. The relationship between the 4th difference and the leading coefficient \( a \) for a polynomial of degree 4 is given by: \[ 24a = \text{constant 4th difference} \] Since the constant 4th difference is 48: \[ 24a = 48 \] To find \( a \), divide both sides by 24: \[ a = \frac{48}{24} = 2 \]
Step 4: Formulate the Polynomial with \( a \)
Now that we have \( a = 2 \), we can start our polynomial: \[ P(n) = 2n^4 + bn^3 + cn^2 + dn + e \]
Step 5: Collect More Information (If Needed)
To find the other coefficients \( b, c, d, \) and \( e \), you typically would need additional information, like specific terms of the sequence or initial conditions. For example, if you know the first few terms of the sequence \( P(1), P(2), \) etc., you can set up equations to solve for these coefficients.
Step 6: Example of Finding Coefficients (Hypothetical)
Let’s say the first four terms of the sequence were:
- \( P(1) = 10 \)
- \( P(2) = 30 \)
- \( P(3) = 70 \)
- \( P(4) = 130 \)
You would plug in these values into the polynomial for \( n = 1, 2, 3, \) and \( 4 \) to get a system of equations:
- For \( n = 1: \) \( 2(1)^4 + b(1)^3 + c(1)^2 + d(1) + e = 10 \)
- For \( n = 2: \) \( 2(2)^4 + b(2)^3 + c(2)^2 + d(2) + e = 30 \)
- For \( n = 3: \) \( 2(3)^4 + b(3)^3 + c(3)^2 + d(3) + e = 70 \)
- For \( n = 4: \) \( 2(4)^4 + b(4)^3 + c(4)^2 + d(4) + e = 130 \)
Step 7: Solve the System of Equations
By solving this system of equations (which might require some algebraic manipulation), you will find values for \( b, c, d, \) and \( e \).
Step 8: Write the Final Polynomial
Once you have determined all the coefficients \( a, b, c, d, \) and \( e \), you can write down the final polynomial \( P(n) \) which will represent the sequence.
Conclusion
In summary, create a polynomial of degree 4 with the first step of determining the leading coefficient from the constant 4th difference, then use any known terms of the sequence to find the other coefficients. This will give you an approximate polynomial expression for your sequence!
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