Explain the steps to construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48.(4 points)

Answer 1

To construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48, follow these steps:

1. Start by writing down the given sequence.

2. Compute the differences between consecutive terms of the sequence.

3. Repeat step 2 until you obtain a sequence of constant differences.

4. Compute the fourth differences of the sequence obtained in step 3.

5. Since the fourth differences are constant and equal to 48, we can construct a polynomial expression of degree 4 (or higher) that has these fourth differences.

6. Use the method of finite differences to find the coefficients of the polynomial expression. This involves constructing a tabular form with the terms, differences, second differences, third differences, and fourth differences. Fill in the gaps in the tabular form using the known information until you have enough equations to solve for the unknown coefficients.

7. Once you have found the coefficients, construct the polynomial expression by adding the terms with their corresponding coefficients. The polynomial expression should have the form: P(x) = ax^4 + bx^3 + cx^2 + dx + e, where a, b, c, d, and e are the coefficients found in step 6.

Note: The polynomial expression obtained may not perfectly represent the given sequence but it will closely approximate it based on the 4th differences.