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

Answer 1

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.