To determine at which differences the given polynomial sequence reaches a constant value, we need to compute successive differences of the sequence.
Given the polynomial:
[tex]\[ a_n = 2n^4 - n^3 \][/tex]
We then compute the first, second, third, and fourth differences as follows:
### First Differences
The first differences are calculated by finding the difference between consecutive terms in the sequence:
[tex]\[ a_{n+1} - a_n \][/tex]
### Second Differences
The second differences are calculated by finding the difference between consecutive first differences:
[tex]\[ (a_{n+1} - a_n) - (a_n - a_{n-1}) \][/tex]
### Third Differences
The third differences are calculated by finding the difference between consecutive second differences:
[tex]\[ [ (a_{n+1} - a_n) - (a_n - a_{n-1}) ] - [ (a_n - a_{n-1}) - (a_{n-1} - a_{n-2}) ] \][/tex]
### Fourth Differences
The fourth differences are calculated by finding the difference between consecutive third differences:
[tex]\[ [ ( (a_{n+1} - a_n) - (a_n - a_{n-1}) ) - ( (a_n - a_{n-1}) - (a_{n-1} - a_{n-2}) ) ] - [ ( (a_n - a_{n-1}) - (a_{n-1} - a_{n-2}) ) - ( (a_{n-1} - a_{n-2}) - (a_{n-2} - a_{n-3}) ) ] \][/tex]
By performing these calculations, we obtain the differences:
- First Differences: [23, 111, 313, 677]
- Second Differences: [88, 202, 364]
- Third Differences: [114, 162]
- Fourth Differences: [48]
Examining these results, we see that the differences reach a constant value at the fourth differences, which is 48.
Therefore, the correct answer is:
4th differences
