To construct a polynomial expression that represents a sequence with constant 4th differences, you can follow a systematic approach. Let's go through the steps together at a level suitable for a 10th grader.
Step 1: Understand the Definition
A sequence with constant 4th differences means that if you look at the differences between consecutive terms of the sequence four times, you will always get the same number. In this case, the constant is 48.
Step 2: Determine the Degree of the Polynomial
Since the 4th differences are constant, the polynomial representing the sequence will be of degree 4. This means it will look something like:
\[ P(n) = an^4 + bn^3 + cn^2 + dn + e \]
where \( a, b, c, d, \) and \( e \) are constants we need to find.
Step 3: Use the Constant 4th Difference
The value of the constant 4th difference can help us determine the leading coefficient \( a \). The relationship between the constant 4th difference and the leading coefficient of a polynomial is:
\( 24a = \text{constant 4th difference} \)
Here, since the constant 4th difference is 48, we can set up the equation:
\[ 24a = 48 \]
Now, solve for \( a \):
\[ a = \frac{48}{24} = 2 \]
So we can write:
\[ P(n) = 2n^4 + bn^3 + cn^2 + dn + e \]
Step 4: Find the Other Coefficients
To find the other coefficients \( b, c, d, \) and \( e \), we normally need some terms from the sequence itself. However, since we don't have specific terms given, let's pretend we have the first few terms of the sequence, such as: \( P(1), P(2), P(3), P(4) \).
Example Terms (Hypothetical):
- \( P(1) = 5 \)
- \( P(2) = 18 \)
- \( P(3) = 37 \)
- \( P(4) = 62 \)
Step 5: Set Up and Solve a System of Equations
Using the values we have, we can set up equations by substituting \( n = 1, 2, 3, \) and \( 4 \):
- \( P(1) = 2(1)^4 + b(1)^3 + c(1)^2 + d(1) + e = 2 + b + c + d + e = 5 \)
- \( P(2) = 2(2)^4 + b(2)^3 + c(2)^2 + d(2) + e = 32 + 8b + 4c + 2d + e = 18 \)
- \( P(3) = 2(3)^4 + b(3)^3 + c(3)^2 + d(3) + e = 162 + 27b + 9c + 3d + e = 37 \)
- \( P(4) = 2(4)^4 + b(4)^3 + c(4)^2 + d(4) + e = 512 + 64b + 16c + 4d + e = 62 \)
Step 6: Solve the Equations
This gives you a system of equations. You will need to solve these equations simultaneously to find the values of \( b, c, d, \) and \( e \).
Step 7: Write the Final Polynomial
Once you find \( b, c, d, \) and \( e \), you can substitute them back into your polynomial form:
\[ P(n) = 2n^4 + bn^3 + cn^2 + dn + e \]
And that's how you find an approximate polynomial expression for a sequence with constant 4th differences of 48! Remember, the specific coefficients depend on the actual sequence terms you have.