Question

At which differences does the following polynomial sequence reach a constant value?

an=2n4−n3

(1 point)
Responses

1st differences
1st differences

2nd differences
2nd differences

3rd differences
3rd differences

4th differences

Answer 1

To determine at which differences the polynomial sequence \( a_n = 2n^4 - n^3 \) reaches a constant value, we can analyze the differences step by step.

1. **Determine the degree of the polynomial**: The polynomial \( a_n = 2n^4 - n^3 \) is a degree 4 polynomial. Generally, for a polynomial of degree \( k \), the \( k \)th differences will reach a constant value.

2. **Calculate the differences**:
- **1st differences**: \( a_{n+1} - a_n \)
- **2nd differences**: \( (a_{n+1} - a_n) - (a_n - a_{n-1}) \)
- **3rd differences**: \( (2nd \ differences) - (2nd \ differences \ for \ n-1) \)
- **4th differences**: \( (3rd \ differences) - (3rd \ differences \ for \ n-1) \)

Since we know from polynomial behavior that:

- The 1st differences of a polynomial of degree 4 will not be constant.
- The 2nd differences will not be constant.
- The 3rd differences will not be constant.
- The 4th differences will be constant.

Thus, for the polynomial \( a_n = 2n^4 - n^3 \), the sequence reaches a constant value at the **4th differences**.

**Final Answer: 4th differences**