sengihantq

2022-10-03

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

Lamar Esparza

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 $r\ne 0$ and $r\ne 1$.
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
${a}_{n}=0\cdot {\left(\frac{1}{2}\right)}^{n}=0$
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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