To create a polynomial representing a sequence with constant fourth differences of 48, you start by calculating the differences between the terms of the sequence. For example, with the sequence 2, 6, 16, 34, 62, you compute the first differences (6-2=4, 16-6=10, 34-16=18, 62-34=28), then the second differences (10-4=6, 18-10=8, 28-18=10), followed by the third differences (8-6=2 and 10-8=2). Finally, you check the fourth difference, which is 2-2=0. A constant fourth difference of 48 indicates that the sequence can be modeled by a fourth-degree polynomial. To find this polynomial, you can use a general form and substitute values from the sequence to create a system of equations, solving for the coefficients to obtain the polynomial expression that approximates the sequence.
simplify "To create a polynomial that represents a sequence with constant fourth differences of 48, you first need to find the differences between the terms of the sequence. For example, if your sequence is 2, 6, 16, 34, 62, you calculate the first differences: 6-2=4, 16-6=10, 34-16=18, and 62-34=28. Next, you find the second differences by subtracting the first differences: 10-4=6, 18-10=8, and 28-18=10. Then, for the third differences, you do the same: 8-6=2 and 10-8=2. Finally, you check the fourth difference, which is 2-2=0. Since the fourth difference is constant and equal to 48, it indicates that the sequence can be modeled by a fourth-degree polynomial. To find the polynomial, you can use the general form
. By substituting the values of and from the sequence into this equation, you can create a system of equations. Solving these equations will give you the coefficients and , resulting in the polynomial expression that approximates the sequence with the specified fourth differences." in to 1 paragraph with no bullet points
1 answer
1 answer