Definition of a geometric sequence

Solution 1:

Ultimately it will depend on what you consider the definition of a geometric series, as you can see by the other answers here. If the definition relies on calculating a common ratio $r = a_{k+1}/a_k$, then you will run into division by $0$.

You can avoid that problem by saying that a geometric sequence is one whose terms obey the property $a_ka_{k+2} = a_{k+1}^2$. Another way is to define the sequence as $a_k = cr^k$.

Personally, I would say that the sequence $0,0,0,\ldots$ is a geometric sequence in the same way that a point is a circle of radius $0$.

Solution 2:

Just to play devil's advocate, here are some theorems we lose if we allow a sequence that ends in zeroes to be geometric:

• A geometric sequence converges if and only if the common factor is in $(-1,1]$. (Counterexample: $0\cdot 2^{n-1}$ does converge).

• A geometric sequence has a sum if and only if the common factor is numerically less than $1$. (Counterexample: $0\cdot 2^{n-1}$ does have a sum).

• If $(a_n)$ and $(b_n)$ are geometric sequences and for some $i$ we have both $a_i=b_i$ and $a_{i+1}=b_{i+1}$, then $a_i=b_i$ for all $i$. (Counterexample: $0,0,0,\ldots$ versus $1,0,0,\ldots$).

Pragmatically it is probably most useful to exclude these sequences from being "geometric" such that the statements of the general properties can be kept simple. If we need to apply the theorems in a situation where the sequence might be degenerate, it is usually very simple to handle degenerate cases by ad-hoc methods.