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Sorry for rather ambiguous title, but I was not sure who to call it otherwise.

So in arithmetic series we have the common ratio $r$ in the form of $+r$. i.e. $2,4,6,8...$ Geotmeric series we have a common ratio in the form $*r$, i.e. $2,6,18,54...$

Do we have a name for a series where the common ratio is of the form ^r. i.e.

$$S=2,4,16,256,65536...$$

here to get to next term we rise the previous term to the power of 2

or $$S=3,27,19683...$$ here we rise to the power of 3

Could we find a nth term of such sequence? a sum of it? all the usual stuff we can apply to arithmetic and geometric series? Does it have a name?

Scavenger23
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1 Answers1

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The $n$th term of this series is $r^{(r^n)}$, we can prove it using simple induction:

For $n=0$, $r^{r^0}=r^1=r$. And if we assume that $a_n=r^{r^n}$, then $a_{n+1}=(r^{r^n})^r=r^{r^n \cdot r}=r^{r^{n+1}}$.

I am not aware of a name for this type of series.

AlephZero
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  • I understand any of those is written as $a(n)=a_{1}$^$(r^n)$, just wanted to ask in general – Scavenger23 Nov 30 '18 at 21:41
  • Ok, I understand. Sorry. – AlephZero Nov 30 '18 at 21:48
  • Oh no problem, again my examples were really ambiguous, cause the first term was the same as the common power. We could also have something like

    $S=5,25,625,390625...$ or $S=64,8,2\sqrt2...$

    ... which of course answers the question about the nth term. I didn't really think it through haha.

     – Scavenger23 Nov 30 '18 at 21:52