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: $$\begin{array}{rlrlrl}n=1:& 1^2+1+1=3n=2:& 2^2+2+1=7n=3:& 3^2+3+1=13n=4:& 4^2+4+1=21n=5:& 5^2+5+1=31\end{array}$$ So the sequence is: $3,7,13,21,31$.

2. Calculate the first differences: $$\begin{array}{rlrlr}7-3& =413-7& =621-13& =831-21& =10\end{array}$$ The first differences are: $4,6,8,10$.

3. Calculate the second differences: $$\begin{array}{rlrl}6-4& =28-6& =210-8& =2\end{array}$$ 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 $2$.