Note that $221 = 13 \times 17$ and therefore $222 = 13 \times 17 + 1$ is in the arithmetic sequence. Furthermore, $222\,222\,221$ is divisible by $13$ and hence $222\,222\,222$ is in the sequence, since
$$222222 = 222 \times 1001 = 222 \times 7 \times 11 \times 13.$$
In fact, any number consisting of $6m + 3$ $2$s will be one more than a multiple of $13$, since the number consisting of $6m$ $2$s will be a multiple of $1001$ and hence a multiple of $13$. Thus there are infinitely many terms in the sequence that only uses the digit $2$.