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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.