Answer:
Explanation:
{\displaystyle {}^{n}x}{}^{n}x, for n = 2, 3, 4, …, showing convergence to the infinitely iterated exponential between the two dots
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.
Under the definition as repeated exponentiation, the notation {\displaystyle {^{n}a}}{\displaystyle {^{n}a}} means {\displaystyle {a^{a^{\cdot ^{\cdot ^{a}}}}}}{\displaystyle {a^{a^{\cdot ^{\cdot ^{a}}}}}}, where n copies of a are iterated via exponentiation, right-to-left, I.e. the application of exponentiation {\displaystyle n-1}n-1 times. n is called the "height" of the function, while a is called the "base," analogous to exponentiation. It would be read as "the nth tetration of a".
Tetration is also defined recursively as
{\displaystyle {^{n}a}:={\begin{cases}1&{\text{if }}n=0\\a^{\left(^{(n-1)}a\right)}&{\text{if }}n>0\end{cases}}}{\displaystyle {^{n}a}:={\begin{cases}1&{\text{if }}n=0\\a^{\left(^{(n-1)}a\right)}&{\text{if }}n>0\end{cases}}},
allowing for attempts to extend tetration to non-natural numbers suc
