Question

Is 0,0,0,... a geometric sequence?

Answer 1

Probably, depending on the definition being used.

Explanation:

In general, a geometric sequence to be one of the form `a_n = a_0r^n` where `a_0` is the initial term and `r` is the common ratio between terms.

In some definitions of a geometric sequence (for example, at the encyclopedia of mathematics) we add a further restriction, dictating that `r!=0` and `r!=1`.
By those definitions, a sequence such as `1, 0, 0, 0, ...` would not be geometric, as it has a common ratio of `0`.

There is one more detail to consider, though. In the given sequence of `0, 0, 0, ...`, we have `a_0 = 0`. In no definition that I have found is there any restriction on `a_0`, and with `a_0=0`, the given sequence could have any common ratio. For example, if we took `r = 1/2` the sequence would look like

`a_n = 0*(1/2)^n = 0`

which does not contradict the definition (note that the definition does not require `r` to be unique).

So, depending on the definition, `0, 0, 0, ...` would probably be considered a geometric sequence.

Still, whether `0, 0, 0, ...` is a geometric sequence or not is likely of little consequence, as the properties and behavior of the sequence are obvious without any further classification.