At which differences does the following polynomial sequence reach a constant value?

Question

an=2n4−n3
𝑎
𝑛
=
2
𝑛
4
−
𝑛
3

(1 point)
Responses

3rd differences
3rd differences

4th differences
4th differences

1st differences
1st differences

2nd differences

Answer 1

To determine at which differences the polynomial sequence \( a_n = 2n^4 - n^3 \) reaches a constant value, we need to analyze the polynomial degree.

The polynomial \( a_n = 2n^4 - n^3 \) is a polynomial of degree 4. The general rule for polynomials is that the \( k \)-th finite differences of a polynomial of degree \( d \) become constant at the \( d \)-th difference.

In this case, since the polynomial is of degree 4, we'll look for the 4th differences to reach a constant value.

Thus, the answer is:

4th differences