Question
The sum of a sequence of consecutive numbers from 1 to n is given by the formula an = x^3 + 2x^2 - 3x + 7. Which level of diferences of this polynomial is a constant value? (1 point)
A. 3rd difference
B. 1st difference
C. 7th difference
D. 2nd difference
A. 3rd difference
B. 1st difference
C. 7th difference
D. 2nd difference
Answers
GPT-4o mini
To determine which level of differences of the polynomial \( a_n = x^3 + 2x^2 - 3x + 7 \) is constant, we can examine the degree of the polynomial.
The polynomial given is of degree 3 (since the highest exponent of \( x \) is 3). For a polynomial of degree \( n \):
- The \( n \)-th differences will be constant.
- The \( (n-1) \)-th differences will form a linear sequence (and thus not be constant).
- The \( (n-2) \)-th differences will form a quadratic sequence, and so on.
For our polynomial of degree 3:
- The **3rd differences** will be constant.
- The **2nd differences** will not be constant (they will be linear).
- The **1st differences** will not be constant (they will be quadratic).
- \( 0 \)-th differences aren't applicable when considering differences since they involve the actual terms themselves.
Therefore, the answer is:
**A. 3rd difference**.
The polynomial given is of degree 3 (since the highest exponent of \( x \) is 3). For a polynomial of degree \( n \):
- The \( n \)-th differences will be constant.
- The \( (n-1) \)-th differences will form a linear sequence (and thus not be constant).
- The \( (n-2) \)-th differences will form a quadratic sequence, and so on.
For our polynomial of degree 3:
- The **3rd differences** will be constant.
- The **2nd differences** will not be constant (they will be linear).
- The **1st differences** will not be constant (they will be quadratic).
- \( 0 \)-th differences aren't applicable when considering differences since they involve the actual terms themselves.
Therefore, the answer is:
**A. 3rd difference**.