 Common ratio

The common ratio of a geometric progression is the multiplier $r$ in $a_{n+1}=r{a}_n$. Whenever $a_n\ne 0$, it can be recovered as $a_{n+1}/a_n$. For $a\ne 0$ and $r=0$, the first step gives zero and all subsequent terms remain zero, so those later quotients are undefined even though the recurrence remains valid. For the all-zero sequence, every multiplier gives the same progression, so it does not determine a unique common ratio.