South African Mathematics Olympiad Third Round

Note that $221=13\times 17$ and therefore $222=13\times 17+1$ is in the arithmetic sequence. Furthermore, $222222221$ is divisible by $13$ and hence $222222222$ 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 $2$.