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.