Question
At which differences does the following polynomial sequence reach a constant value?
(1 point)
an = 2n −
4 n
3
1st differences
3rd differences
4th differences
2nd differences
(1 point)
an = 2n −
4 n
3
1st differences
3rd differences
4th differences
2nd differences
Answers
GPT-4o mini
To determine at which differences the polynomial sequence given by \( a_n = 2n - 4n^3 \) reaches a constant value, we first need to note the degree of the polynomial.
1. **Identify the degree of the polynomial**: The polynomial \( a_n = 2n - 4n^3 \) is a cubic polynomial, as the highest power of \( n \) is 3.
2. **Calulate the differences**:
- **1st Differences**: The first differences \( \Delta a_n = a_{n+1} - a_n \).
- **2nd Differences**: The second differences \( \Delta^2 a_n = \Delta a_{n+1} - \Delta a_n \).
- **3rd Differences**: The third differences \( \Delta^3 a_n = \Delta^2 a_{n+1} - \Delta^2 a_n \).
- **4th Differences**: The fourth differences \( \Delta^4 a_n = \Delta^3 a_{n+1} - \Delta^3 a_n \).
3. **Determine where it becomes constant**: For a polynomial of degree \( k \):
- The \( k^{th} \) differences are constant.
- All higher differences (i.e., \( (k+1)^{th} \) and beyond) will be zero.
Since our polynomial has a degree of 3, the differences are as follows:
- The **1st differences** are not constant (because the degree is 3).
- The **2nd differences** are not constant (still not reduced to a constant value).
- The **3rd differences** will be constant.
- The **4th differences** will be zero (and therefore constant).
Thus, the polynomial \( a_n = 2n - 4n^3 \) reaches a constant value at its **3rd differences**.
1. **Identify the degree of the polynomial**: The polynomial \( a_n = 2n - 4n^3 \) is a cubic polynomial, as the highest power of \( n \) is 3.
2. **Calulate the differences**:
- **1st Differences**: The first differences \( \Delta a_n = a_{n+1} - a_n \).
- **2nd Differences**: The second differences \( \Delta^2 a_n = \Delta a_{n+1} - \Delta a_n \).
- **3rd Differences**: The third differences \( \Delta^3 a_n = \Delta^2 a_{n+1} - \Delta^2 a_n \).
- **4th Differences**: The fourth differences \( \Delta^4 a_n = \Delta^3 a_{n+1} - \Delta^3 a_n \).
3. **Determine where it becomes constant**: For a polynomial of degree \( k \):
- The \( k^{th} \) differences are constant.
- All higher differences (i.e., \( (k+1)^{th} \) and beyond) will be zero.
Since our polynomial has a degree of 3, the differences are as follows:
- The **1st differences** are not constant (because the degree is 3).
- The **2nd differences** are not constant (still not reduced to a constant value).
- The **3rd differences** will be constant.
- The **4th differences** will be zero (and therefore constant).
Thus, the polynomial \( a_n = 2n - 4n^3 \) reaches a constant value at its **3rd differences**.