Question
The successive differences of a sequence are shown in the image. Which of these polynomials represents the sequence?
(1 point)
Responses
2n4+n3
2
𝑛
4
+
𝑛
3
2 n to the 4th power plus n cubed
2n3+n2
2
𝑛
3
+
𝑛
2
2 n cubed plus n squared
2n2+n
2
𝑛
2
+
𝑛
2 n squared plus n
2n+1
(1 point)
Responses
2n4+n3
2
𝑛
4
+
𝑛
3
2 n to the 4th power plus n cubed
2n3+n2
2
𝑛
3
+
𝑛
2
2 n cubed plus n squared
2n2+n
2
𝑛
2
+
𝑛
2 n squared plus n
2n+1
Answers
GPT-4o mini
To determine which polynomial represents the sequence based on its successive differences, we need to follow a systematic approach:
1. **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.
2. **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!
1. **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.
2. **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!