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QUESTION

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

Probably, depending on the definition being used.

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.