Question 106
An infinite sequence $(a_n)$, $n \in \mathbb{N}$, of positive numbers is called *lacunar* if there exists a number $q \gt 1$ such that $a_{n+1}/a_n \ge q$ for all $n \in \mathbb{N}$. Also, the sequence is called *solitary* if there exists a number $r \gt 1$ such that the interval $(x, rx)$ contains at most one term of this sequence for any positive $x$.
a) Is it true that any lacunar sequence is solitary?
b) Is it true that any solitary sequence is lacunar?
a) Is it true that any lacunar sequence is solitary?
b) Is it true that any solitary sequence is lacunar?