Question

which polynomial sequence shows that the terms of the 2nd differences are cibsrabt

Answer 1

To understand the concept of second differences and how they relate to polynomial sequences, let's first clarify what first and second differences are:

1. **First Differences**: Given a sequence of terms \( a_0, a_1, a_2, \ldots \), the first differences are obtained by subtracting consecutive terms:
\[
\Delta a_n = a_{n+1} - a_n
\]

2. **Second Differences**: The second differences are the differences of the first differences:
\[
\Delta^2 a_n = \Delta a_{n+1} - \Delta a_n
\]

Now, for polynomial sequences:

- **Linear polynomials (degree 1)** have constant first differences, which implies that their second differences are zero.
- **Quadratic polynomials (degree 2)** have constant second differences.
- **Cubic polynomials (degree 3)** have constant third differences.

The term "cibsrabt" seems to be a typographical error. If you meant "constant," then it is important to note that for a polynomial of degree 2, the second differences are constant.

### Example of a Quadratic Polynomial

Consider the quadratic polynomial:
\[
a_n = n^2
\]

Calculating the first differences:
\[
\Delta a_n = (n+1)^2 - n^2 = 2n + 1
\]

Now, calculating the second differences:
\[
\Delta^2 a_n = (2(n+1) + 1) - (2n + 1) = 2
\]
The second differences are constant and equal to 2.

### Conclusion

If you are looking for a polynomial sequence where the second differences are constant, then quadratic polynomials (such as \( a_n = n^2 \)) fit this criterion. If you have a different polynomial degree in mind or a specific sequence context, please provide more detail, and I can assist you further!