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

Dec 15, 2015

Probably, depending on the definition being used.

#### Explanation:

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 $1 , 0 , 0 , 0 , \ldots$ 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 $0 , 0 , 0 , \ldots$, 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, $0 , 0 , 0 , \ldots$ would probably be considered a geometric sequence.

Still, whether $0 , 0 , 0 , \ldots$ 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.