"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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