Question

What are the 2nd differences of the sequence from the following polynomial? n^2+n+1

Answer 1

To find the 2nd differences of the sequence generated by the polynomial \( n^2 + n + 1 \), we first need to evaluate the polynomial for several values of \( n \) and then calculate the first and second differences.

1. **Calculate the values of the polynomial:**
\[
\begin{align*}
n = 1: & \quad 1^2 + 1 + 1 = 3 \\
n = 2: & \quad 2^2 + 2 + 1 = 7 \\
n = 3: & \quad 3^2 + 3 + 1 = 13 \\
n = 4: & \quad 4^2 + 4 + 1 = 21 \\
n = 5: & \quad 5^2 + 5 + 1 = 31 \\
\end{align*}
\]
So the sequence is: \( 3, 7, 13, 21, 31 \).

2. **Calculate the first differences:**
\[
\begin{align*}
7 - 3 & = 4 \\
13 - 7 & = 6 \\
21 - 13 & = 8 \\
31 - 21 & = 10 \\
\end{align*}
\]
The first differences are: \( 4, 6, 8, 10 \).

3. **Calculate the second differences:**
\[
\begin{align*}
6 - 4 & = 2 \\
8 - 6 & = 2 \\
10 - 8 & = 2 \\
\end{align*}
\]
The second differences are: \( 2, 2, 2 \).

Thus, the second differences of the sequence generated by the polynomial \( n^2 + n + 1 \) are constant and equal to \( 2 \).