Add a reciprocal, repeat

Algebra Level 5

Let \( a_n \) be a sequence of positive real numbers defined by \( a_ 1 = 1 \) and

\[ a_{n+1} = a_n + \frac{1}{a_n} \text{ for } n\geq 1. \]

Evaluate \( \lfloor a_{100000} \rfloor \).

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