This is somewhat related to Erdős conjecture on arithmetic progressions
Is there an infinite sequence of positive integers $a_n$ s.t. $\sum_{n=1}^\infty{\frac{1}{a_n}}$ converges and $a_n$ contains arbitrary long arithmetic progressions?
If one allows negative integers a solution is $a_n=(-1)^n n$
$\{a_n\}$
and a very fastly growing integer sequence$\{b_n\}$
. In succession, add $b_1$ to all the terms of $a_n$ after the first, add $b_2$ to all the terms after the third, ..., add $b_r$ to all the terms after the $\binom{r+1}{2}$-th. If $b_n$ is at least (roughly) $n^4$, say $b_n=((n!)!)!$, this should work. $\endgroup$