To construct an approximate polynomial expression for a sequence that has constant fourth differences of 48, I started by noting that because the fourth difference is constant, the polynomial must be of degree 4. Then I determined the leading coefficient, which is connected to the fourth difference. Since the fourth difference is 48, I divided it by 24 (the factorial of 4) to find the leading coefficient of the polynomial; this gave me a coefficient of 2. Next, I wrote the polynomial in its general form as \P(n) = 2n^4 + an^3 + bn^2 + cn + d\), where $a$, $b$, $c$, and $d$are constants that I still need to determine. To find these constants, I used initial conditions or values from the given sequence to create a system of equations. This allowed me to solve for $a$, $b$, $c$, and $d$.Finally, after substituting these constants back into the polynomial, I have the desired expression that approximates the original sequence.

Question

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GPT-4o mini · Answer

To create an approximate polynomial expression for a sequence with constant fourth differences of 48, I recognized that a polynomial of degree 4 would be suitable due to the constant fourth differences. Knowing that the fourth difference is consistent, the polynomial's leading coefficient can be determined by dividing this value by 24, which is the factorial of 4. In this case, dividing 48 by 24 gives a leading coefficient of 2.

I then expressed the polynomial in a general form: $P(n) = 2n^4 + an^3 + bn^2 + cn + d$, where $a$, $b$, $c$, and $d$ are constants that need to be determined. To find these constants, I utilized initial values from the given sequence to set up a system of equations. By solving this system, I was able to determine the values of $a$, $b$, $c$, and $d$. After substituting these constants back into the polynomial, I arrived at the desired expression that approximates the original sequence.