Carl's Arithmetic Sequence
An arithmetic sequence of positive integers has $n \geq 3$ terms, initial term $a$, and common difference $d > 1$. Carl wrote down all the terms in this sequence correctly except for one term, which was off by $1$. The sum of the terms he wrote was $222$. What is $a+d+n$?
\[
\textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 20 \qquad
\textbf{(C)}\ 22 \qquad
\textbf{(D)}\ 28 \qquad
\textbf{(E)}\ 26
\]