Which of the following defines a geometric sequence?

As others have noted, it all depends on how precisely one defines the term geometric sequence. One can define anything one likes; for instance Wikipedia gives the definition that the general term at position $n$ (starting from $0$) should be of the form $ar^n$ with the rather ridiculous condition that $r\neq0$ though $a$ is unconstrained; that would make $0,0,\ldots$ into a geometric series of undetermined ratio $r\neq0$ (taking $a=0$), but $1,0,0,\ldots$ would not be a geometric series (here the ratio would clearly have to be $0$, but that was forbidden).

Part of the problem here is that one tries to define the unqualified notion "geometric progression" (or sequence), while in practice one most often deals with geometric progressions of specified ratio $r$ (in which case the general term is simply $ar^n$ where $a$ is the initial term); this shifts the focus from generating a geometric progression (which really never poses any problem) to recognising a given sequence as being geometric (this leads to considering ratios of successive terms, which is problematic if they are $0$; it still does not explain why one should forbid $r$ to be $0$, but not $a$).

To take some distance from the rather uninspiring "I found a source that says this" approach, let me view geometric sequences as an instance of anther notion, that of eigenvector. In the vector space of infinite sequences, the geometric sequences with ratio $r$ are precisely the eigenvectors of the "guillotine operator", which removes the initial term of a sequence and shifts the other terms to fill its place, for the eigenvalue$~r$. From this perspective, one sees why the zero sequence should have a special status; normally the zero vector is explicitly forbidden as eigenvector (because it would be so for any value of$~\lambda$, whether or not it is an eigenvalue), although it is included in any eigenspace. For the exact same reason allowing the zero sequence as a geometric sequence is somewhat problematic (no unique ratio can be determined for it) but it is OK (in fact necessary) to include the zero sequence in the subspace of geometric sequences with given ratio$~r$.

Note that viewing geometric sequences as eigenvectors is quite natural; they are used in this role when solving linear recurrence relations with constant coefficients.

Which of the following defined as a geometric sequence?

How do you identify that a sequence is a geometric sequence?

What defines a geometric series?

What is a geometric sequence example?