To find out between which two consecutive terms the first difference equals 2 in the sequence defined by \( a_n = n^2 + n + 1 \), we first need to calculate the first difference \( a_{n+1} - a_n \).
The sequence's elements are given by:
\[
a_n = n^2 + n + 1
\]
\[
a_{n+1} = (n+1)^2 + (n+1) + 1 = n^2 + 2n + 1 + n + 1 + 1 = n^2 + 3n + 3
\]
Now, we find the first difference:
\[
a_{n+1} - a_n = (n^2 + 3n + 3) - (n^2 + n + 1)
\]
\[
= n^2 + 3n + 3 -