To determine the degree of a polynomial sequence such that its 6th differences are a constant value, we need to understand the relationship between the differences of a polynomial sequence and its degree.
1. Understanding Differences in Polynomial Sequences:
- The first differences of a sequence provide the differences between consecutive terms.
- The second differences are the differences of the first differences, and so on.
2. General Rule:
- For a polynomial of degree [tex]\( n \)[/tex], the [tex]\( (n+1) \)[/tex]-th differences are constant.
- For example, if you have a polynomial of degree [tex]\( 1 \)[/tex] (a linear function), then the first differences are constant.
- If you have a polynomial of degree [tex]\( 2 \)[/tex] (a quadratic function), then the second differences are constant.
- Extending this pattern, for a polynomial of degree [tex]\( 3 \)[/tex], the third differences are constant, and so forth.
3. Solving for the Given Condition:
- We need the 6th differences to be constant. According to the rule mentioned above, if the 6th differences are constant, the degree of the polynomial should be one less than 6+1.
4. Calculating the Degree:
- More formally, if the [tex]\( k \)[/tex]-th differences of a polynomial are constant, the polynomial must be of degree [tex]\( k-1 \)[/tex].
Given:
- The 6th differences are constant.
Therefore, the polynomial sequence must be of degree:
[tex]\[ 6 - 1 = 6 \][/tex]
Hence, the degree of the polynomial sequence should be 6 for its 6th differences to be a constant value.
