Let's solve the question step-by-step.
1. Understanding Differences in Polynomial Sequences:
- For a polynomial sequence [tex]\( a(n) \)[/tex], the sequence values are represented by the polynomial.
- The differences between consecutive terms of the sequence provide insights into the polynomial's behavior and degree.
2. Properties of Differences:
- The first differences of the sequence [tex]\( a(n) \)[/tex] are the differences between each consecutive term.
- The second differences are the differences of the first differences, and this process continues for higher-order differences.
3. Degree of Polynomials and Constant Differences:
- A polynomial of degree [tex]\( n \)[/tex] has differences that become constant at the [tex]\( n \)[/tex]-th difference level.
- For example, a polynomial of degree 1 (a linear polynomial like [tex]\( ax + b \)[/tex]) has constant first differences.
- More generally, a polynomial of degree [tex]\( n \)[/tex] will have constant [tex]\( n \)[/tex]-th differences.
4. Specific Case: 6th Differences:
- We are asked to find the degree of the polynomial such that the 6th differences are constant.
- By the properties noted above, if the 6th differences of a sequence are constant, the polynomial must be of degree 6.
Therefore, the degree of the polynomial sequence for its 6th differences to be a constant value is:
[tex]\[ \boxed{6} \][/tex]
