Question
Use the image to answer the 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 An+B upper A n plus upper B 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
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine the standard form of the polynomial expression based on the provided information about the successive differences, we can follow these steps:
1. **Identify the Order of Differences**: The number of times we need to take the difference until we reach a constant tells us the degree of the polynomial. If the successive differences reach a constant after the \(k^{th}\) difference, then the original polynomial is of degree \(k\).
2. **Selecting the Correct Polynomial**:
- If the first differences (differences taken once) are constant, the polynomial is of degree 1, which corresponds to the form \(An + B\).
- If the second differences are constant, the polynomial is of degree 2, which corresponds to the form \(An^2 + Bn + C\).
- If the third differences are constant, the polynomial is of degree 3, which corresponds to the form \(An^3 + Bn^2 + Cn + D\).
- If the fourth differences are constant, the polynomial is of degree 4, which corresponds to the form \(An^4 + Bn^3 + Cn^2 + Dn + E\).
3. **Conclusion**: Without seeing the image, I recommend you check the levels of the differences given. Based on what you see, you can select the appropriate option:
- If the first differences are constant, the answer is \(An + B\).
- If the second differences are constant, the answer is \(An^2 + Bn + C\).
- If the third differences are constant, the answer is \(An^3 + Bn^2 + Cn + D\).
- If the fourth differences are constant, the answer is \(An^4 + Bn^3 + Cn^2 + Dn + E\).
Please identify at which level the differences stabilize and use that to select the correct polynomial expression.
1. **Identify the Order of Differences**: The number of times we need to take the difference until we reach a constant tells us the degree of the polynomial. If the successive differences reach a constant after the \(k^{th}\) difference, then the original polynomial is of degree \(k\).
2. **Selecting the Correct Polynomial**:
- If the first differences (differences taken once) are constant, the polynomial is of degree 1, which corresponds to the form \(An + B\).
- If the second differences are constant, the polynomial is of degree 2, which corresponds to the form \(An^2 + Bn + C\).
- If the third differences are constant, the polynomial is of degree 3, which corresponds to the form \(An^3 + Bn^2 + Cn + D\).
- If the fourth differences are constant, the polynomial is of degree 4, which corresponds to the form \(An^4 + Bn^3 + Cn^2 + Dn + E\).
3. **Conclusion**: Without seeing the image, I recommend you check the levels of the differences given. Based on what you see, you can select the appropriate option:
- If the first differences are constant, the answer is \(An + B\).
- If the second differences are constant, the answer is \(An^2 + Bn + C\).
- If the third differences are constant, the answer is \(An^3 + Bn^2 + Cn + D\).
- If the fourth differences are constant, the answer is \(An^4 + Bn^3 + Cn^2 + Dn + E\).
Please identify at which level the differences stabilize and use that to select the correct polynomial expression.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.