[Math] the definition of a geometric progression

Question

If the first term in our geometric progression (GP) is $k$, and the common ratio is 0, then our sequence is $\{k, 0, 0, 0, 0,\ldots\}$. Is there anything wrong with this statement?

So, is $\{0, 0, 0,\ldots\}$ a GP?

I have googled for a definition of GP, but wikipedia (which I am skeptical about) is the only link with a definition. (uncited)

Answer 1

From Wikipedia: Geometric Progression Note the very last line.

In mathematics, 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. ... The sum of the terms of a geometric progression, or of an initial segment of a geometric progression, is known as a geometric series.

Thus, the general form of a geometric sequence is

$$a,\, ar,\, ar^2,\ ar^3,\ ar^4,\ \ldots$$

and that of a geometric series is

$$a + ar + ar^2 + ar^3 + ar^4 + \cdots$$

where $r \neq 0$ is the common ratio and $a$ is a scale factor, equal to the sequence's start value. [bold face mine]

Hence, to answer your question, NO, in a geometric progression, the common ratio $r$ cannot be zero.

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See also: Encyclopedia of Mathematics: Geometric Progression

A sequence of numbers each one of which is equal to the preceding one multiplied by a number $q\neq 0$ ...

There's an additional qualification there that $q \;\;\text{or r} \;\;\neq 1$.