Question
The successive differences of a sequence are shown in the image. What is the standard form of its related polynomial expression?
(1 point)
Responses
An3+Bn2+Cn+D
upper A n cubed plus upper B n squared plus upper C n plus upper D
An4+Bn3+Cn2+Dn+E
upper A n superscript 4 baseline plus upper B n cubed plus upper C n squared plus upper D n plus E
An2+Bn+C
upper A n squared plus upper B n plus upper C
An+B
(1 point)
Responses
An3+Bn2+Cn+D
upper A n cubed plus upper B n squared plus upper C n plus upper D
An4+Bn3+Cn2+Dn+E
upper A n superscript 4 baseline plus upper B n cubed plus upper C n squared plus upper D n plus E
An2+Bn+C
upper A n squared plus upper B n plus upper C
An+B
Answers
GPT-4o mini
To determine the standard form of the polynomial expression related to the given sequence based on its successive differences, we need to analyze the degree of the polynomial.
1. If the first differences of the sequence are constant, the sequence can be represented by a polynomial of degree 1, which would be of the form \(An + B\).
2. If the second differences are constant, then the sequence can be represented by a polynomial of degree 2, which would be of the form \(An^2 + Bn + C\).
3. If the third differences are constant, then it can be represented by a polynomial of degree 3, of the form \(An^3 + Bn^2 + Cn + D\).
4. If the fourth differences are constant, it would require a polynomial of degree 4, of the form \(An^4 + Bn^3 + Cn^2 + Dn + E\).
You would need to look at the information regarding the successive differences in the image to determine which case applies. Without visible information on the differences, I can’t conclude which specific polynomial form is correct.
However, if you find that the constant difference corresponds to the first, second, third, or fourth differences, you can select the appropriate polynomial expression based on the explanation above.
1. If the first differences of the sequence are constant, the sequence can be represented by a polynomial of degree 1, which would be of the form \(An + B\).
2. If the second differences are constant, then the sequence can be represented by a polynomial of degree 2, which would be of the form \(An^2 + Bn + C\).
3. If the third differences are constant, then it can be represented by a polynomial of degree 3, of the form \(An^3 + Bn^2 + Cn + D\).
4. If the fourth differences are constant, it would require a polynomial of degree 4, of the form \(An^4 + Bn^3 + Cn^2 + Dn + E\).
You would need to look at the information regarding the successive differences in the image to determine which case applies. Without visible information on the differences, I can’t conclude which specific polynomial form is correct.
However, if you find that the constant difference corresponds to the first, second, third, or fourth differences, you can select the appropriate polynomial expression based on the explanation above.