At which differences does the following polynomial sequence reach a constant value? an=2n4−n3 (1 point) Responses 2nd differences 2nd differences 3rd differences 3rd differences 4th differences 4th differences 1st differences 1st differences

Answer 1

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

The given polynomial \( a_n \) is of degree 4, which means its highest power of \( n \) is \( n^4 \).

For any polynomial of degree \( k \), the \( (k+1) \)-th differences will be constant. Therefore, since our polynomial is of degree 4, we need to consider the 5th differences:

• 1st differences are not constant.
• 2nd differences are not constant.
• 3rd differences are not constant.
• 4th differences are not constant.
• 5th differences will be constant.

Thus, the polynomial sequence \( a_n = 2n^4 - n^3 \) reaches a constant value at the 5th differences.