To construct an approximate polynomial expression for a sequence that has constant fourth differences of 48, I start by noting that because the fourth difference is constant, the polynomial must be of degree 4. Then I determine 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 would use initial conditions or values from the given sequence to create a system of equations. This allows 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

To construct an approximate polynomial expression for a sequence that has constant fourth differences of 48, I start by noting that because the fourth difference is constant, the polynomial must be of degree 4. Then I determine 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 would use initial conditions or values from the given sequence to create a system of equations. This allows 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.

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GPT-4o mini · Answer

To construct an approximate polynomial expression for a sequence that has constant fourth differences of 48, I note that because the fourth difference is constant, the polynomial must be of degree 4. I determine the leading coefficient, which is connected to the fourth difference. Since the fourth difference is 48, I divide it by 24 (the factorial of 4) to find the leading coefficient of the polynomial; this gives me a coefficient of 2. Next, I write 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 would use initial conditions or values from the given sequence to create a system of equations. This allows me to solve for $ a $, $ b $, $ c $, and $ d $. Finally, after substituting these constants back into the polynomial, I will have the desired expression that approximates the original sequence.