# "Infinitely Nested" Tree-Log

Calculus Level 4

Given that $$n > 1$$, $$a_1 > 1$$ and an odd positive integer $$b$$, let:

$\large a_n = b\log_{10}(a_1 + a_{n-1}).$

If the following conditions are true:

• $$b$$ and $$\displaystyle \lim_{n \to \infty} a_n$$ can be arbitrarily chosen.
• $$\displaystyle\lim_{n \to \infty} a_n$$ exists and is finite.
• $$\displaystyle\lim_{n \to \infty} a_n$$ is a positive integer.

What is the least possible value of $$a_1$$ if it must be a perfect square?

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