Is it ok for 'r' to be negative in geometric series?

Question

$6+x+y+z+96 ......$ is a geometric series. Here we need to find the value of $x$. Before doing that we need to find the value of $r$. Here, $a=6$, $ar^4=96$

Now, $r^4=16$, then $r=\pm 2$. But, a group of teacher are saying that here $r$ can't be negative while others are saying that it is okay for $r$ to be negative here. The people who are saying that here $r$ can't be negative, they are giving the logic that as here $a$ is positive, $r$ has no chance to be negative.

But we have seen a lot of geometric series like that. What do you say? Can't $r$ be negative here? Please help me with your logic. Thanks in advance.

Answer 1

tl; dr: In the circumstances, I'd assume the ratio is positive.

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The comments are mathematically correct that a ratio in a geometric series need not be positive.

That said, in the context of a finite geometric series, as is the case here, it would be (at least a little) anomalous if either the initial or final term were anything but a positive real number, and it would be anomalous if the ratio were anything but a positive real number.

By way of evidence, think of

• The concept of geometric means;
• A "geometric subdivision" of a positive real interval $[a, b]$, whose subintervals' lengths are in some ratio $\sqrt[n]{b/a}$;
• The way the neck on a stringed instrument is divided into frets.

To me this is not a question of mathematics, but of (mathematical) cultural expectations. But because the question concerns a potential ambiguity in a basic mathematical concept, I think it's a good question for this site.