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$6+x+y+z+96 ......$ 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.

Vedant Chourey
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Nur Mohammad Khan Othoy
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    $a$ and $r$ can be any values, though you need $|r|<1$ if you want the infinite sum to converge – Henry Nov 18 '22 at 15:25
  • It is. If you have a geometric series in infinitely many terms, then it is only valid if the magnitude is less than one. This can be generalised beyond the real line – FShrike Nov 18 '22 at 15:25
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    $r$ could be complex as well. It is perfectly allowed to call a sequence like $1,2i,-4,-8i,16,\dots$ a geometric sequence. If you are in a scenario where you want only positive real terms, then you can say so... – JMoravitz Nov 18 '22 at 15:26
  • Does this sequence anyhow shows that its terms have to positive? – Nur Mohammad Khan Othoy Nov 18 '22 at 15:29
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    With what you have written by itself, no... nothing suggests with what you have written that the terms need to all be positive reals... however we don't know how it was originally phrased to you. You could have written something differently than the original phrasing. Maybe it was said that "$x,y,z$ are positive real numbers such that $6+x+y+z+96+\dots$ forms a geometric series" or perhaps something else was lost in translation. – JMoravitz Nov 18 '22 at 15:33
  • $r$ can be both negative and positive, the given conditions, that is, $6+x+y+z+96$, is satisfied by both unless it is stated that $x$ cannot be negative. Now, as others have pointed out, $r$ only needs to be less than one (not necessarily negative) if you want to find the sum of an infinite series that is convergent – 冥王 Hades Nov 18 '22 at 15:41

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

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.

Andrew D. Hwang
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    I wish you'd be more emphatic in telling the OP that mathematically a finite geometric series can involve any value of $r$. – hardmath Nov 23 '22 at 00:23