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

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. Let's say the sequence is {a1, a2, a3, a4, ...}.
2. Calculate the differences between consecutive terms. This will give you a new sequence, let's call it {d1, d2, d3, d4, ...}, where d1 = a2 - a1, d2 = a3 - a2, d3 = a4 - a3, and so on.
3. Calculate the differences between consecutive terms of the new sequence. This will give you a second new sequence, let's call it {d'1, d'2, d'3, d'4, ...}, where d'1 = d2 - d1, d'2 = d3 - d2, d'3 = d4 - d3, and so on.
4. Repeat this process two more times, calculating the differences between consecutive terms of the previous sequence each time. This will give you a total of four sequences, {d1, d2, d3, d4, ...}, {d'1, d'2, d'3, d'4, ...}, {d''1, d''2, d''3, d''4, ...}, and {d'''1, d'''2, d'''3, d'''4, ...}.
5. Observe that the constant differences between consecutive terms in the fourth sequence are of value 48. This indicates that the original sequence is likely to be approximated by a polynomial of degree 4.
6. To find the approximate polynomial, we start with a general form: P(x) = ax^4 + bx^3 + cx^2 + dx + e, where a, b, c, d, and e are constants to be determined.
7. Substitute the indices of the original sequence into P(x) and set up a system of equations by equating the polynomial expression with the corresponding terms from the original sequence.
8. Solve the system of equations to find the values of a, b, c, d, and e, which will allow you to construct the approximate polynomial expression to represent the sequence with constant 4th differences of 48.
9. Finally, write down the found polynomial expression.

Note: The specific values of the sequence will determine the exact coefficients of the polynomial expression.