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

Question

Lamar Esparza · Accepted Answer

In general, a geometric sequence to be one of the form

a_{n} = a_{0} r^{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 \neq 0

and

r \neq 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 = \frac{1}{2}

the sequence would look like

a_{n} = 0 \cdot {(\frac{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

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

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