Question

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

an=2n4−n3

(1 point)
Responses

2nd differences
2nd differences

1st differences
1st differences

3rd differences
3rd differences

4th differences

Answer 1

To determine at which differences the polynomial sequence defined by \( a_n = 2n^4 - n^3 \) reaches a constant value, we should look at the degree of the polynomial.

The given polynomial \( a_n \) is a 4th degree polynomial (since the highest power of \( n \) is 4).

In general, for a polynomial of degree \( k \):

- The **1st differences** (the differences between consecutive terms) are typically a polynomial of degree \( k-1 \).
- The **2nd differences** are typically a polynomial of degree \( k-2 \).
- The **3rd differences** are typically a polynomial of degree \( k-3 \).
- The **4th differences** will be constant for a polynomial of degree \( k \).

Since \( a_n \) is a degree 4 polynomial, its 4th differences will be constant, meaning:

**The polynomial sequence \( a_n = 2n^4 - n^3 \) reaches a constant value at the 4th differences.**

Hence, the correct response is:

**4th differences**.