To determine which polynomial represents the sequence based on its successive differences, we need to follow a systematic approach:
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Identify the Degree of the Polynomial: The degree of the polynomial can be inferred from how many times we must take differences until we get a constant value. If you take the first differences and they form a linear sequence, the polynomial is of degree 1. If the second differences are constant, it's degree 2, and so forth.
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Calculate Differences: If you can provide the actual values or the initial sequence, I can help calculate the differences.
In the absence of actual difference data, I can infer from the options given, if we assume that they correspond to different degrees of sequences:
- Degree 4 Polynomial (
2n^4 + n^3) is the highest and would yield increasingly complex differences. - Degree 3 Polynomial (
2n^3 + n^2) would yield non-linear first differences. - Degree 2 Polynomial (
2n^2 + n) would yield constant second differences. - Degree 1 Polynomial (
2n + 1) would yield constant first differences.
If the differences provided are increasing in complexity (say you have differences that become constant after four iterations), you should probably select 2n^4 + n^3.
If you know how many iterations of differences were constant, please provide that, and I can give a precise recommendation!