To determine at which differences the polynomial sequence [tex]\(a_n = 2n^4 - n^3\)[/tex] reaches a constant value, we will need to calculate the first, second, third, and fourth differences of the polynomial sequence. We start by taking derivatives of the polynomial.
1. First difference (first derivative of [tex]\(a_n\)[/tex]):
[tex]\[ a_n = 2n^4 - n^3 \][/tex]
[tex]\[ \frac{d}{dn}(2n^4 - n^3) = 8n^3 - 3n^2 \][/tex]
Therefore, the first difference is [tex]\(8n^3 - 3n^2\)[/tex].
2. Second difference (second derivative of [tex]\(a_n\)[/tex]):
[tex]\[ \frac{d}{dn}(8n^3 - 3n^2) = 24n^2 - 6n \][/tex]
Therefore, the second difference is [tex]\(24n^2 - 6n\)[/tex].
3. Third difference (third derivative of [tex]\(a_n\)[/tex]):
[tex]\[ \frac{d}{dn}(24n^2 - 6n) = 48n - 6 \][/tex]
Therefore, the third difference is [tex]\(48n - 6\)[/tex].
4. Fourth difference (fourth derivative of [tex]\(a_n\)[/tex]):
[tex]\[ \frac{d}{dn}(48n - 6) = 48 \][/tex]
Therefore, the fourth difference is a constant value, [tex]\(48\)[/tex].
Based on the above calculations, the polynomial sequence [tex]\(a_n = 2n^4 - n^3\)[/tex] reaches a constant value at the fourth differences. Thus, the correct answer is:
- 4th differences
