sengihantq

2022-10-03

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

Lamar Esparza

Beginner2022-10-04Added 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 $r\ne 0$ 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

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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