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

sengihantq 2022-10-03 Answered
Is 0,0,0,... a geometric sequence?
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Answers (1)

Lamar Esparza
Answered 2022-10-04 Author has 8 answers
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 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
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