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.

Rewrite this grammar correctly

Answer 1

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.