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

1 Answer
Dec 15, 2015

Answer:

Probably, depending on the definition being used.

Explanation:

In general, a geometric sequence to be one of the form #a_n = a_0r^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.