Infinite Recursive Logarithm Sequence

Calculus Level 1

A recursive sequence is defined as a0=πa_0=\pi and an=logan127.a_n=\log_{a_{n-1}}{27}.

Given that limnan\displaystyle\lim_{n\to\infty}{a_n} exists, what is limnan?\displaystyle\lim_{n\to\infty}{a_n}?


Bonus: For what values of a0a_0 does limnan\displaystyle\lim_{n\to\infty}{a_n} exist?

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