From wiki, the definition of a Geometric Progression is seen as
a geometric progression, also known as a geometric sequence, is a
sequence of numbers where each term after the first is found by
multiplying the previous one by a fixed, non-zero number called the
common ratio. For example, the sequence 2, 6, 18, 54, … is a
geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25,
… is a geometric sequence with common ratio 1/2.
(1) It is clearly mentioned that common ratio cannot be zero. That means,
$8,0,0,0,\cdots$ is not a valid Geometric progression because common ratio is zero. Is my understanding right?
(2) But this does not rule out the possibility of having first term zero. If first term is zero, whatever be the common ratio, all other terms will be zero.
eg: $0,0,0,0,\cdots$. So this must be a valid geometric progression. Is my understanding right?
Best Answer
Pulling what Bungo said, it doesn't actually matter. If you start with $0$, or the common ratio is $0$, or any other trivial point, the math works out just fine, so to me, it is a matter of not caring because it has no actually impact on anything.