Robert Israel previously confirmed that for every finite binary string, there is necessarily some power of 3 that, when expressed in binary, begins with that string.
Does this imply that the same is true for an ordered set of binary strings and sequential powers of 3? That is, for every finite set $\{a_1, a_2, a_3 ...\}$ of finite binary strings there is some $n$ such that each value $3^n, 3^{n+1}, 3^{n+2}$ starts with the corresponding string?
Added: it seems to me that it cannot be true for arbitrary binary strings and sequential powers of 3, because for any number $a$ whose binary representation starts with $100$, $3*a$ cannot start with $11$. But maybe it is still true for a series like $3^n, 3^{n+k}, 3^{n+2k}$?
