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?
Best Answer
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.