Add a reciprocal, repeat

Algebra Level 5

Let an a_n be a sequence of positive real numbers defined by a1=1 a_ 1 = 1 and

an+1=an+1an for n1. a_{n+1} = a_n + \frac{1}{a_n} \text{ for } n\geq 1.

Evaluate a100000 \lfloor a_{100000} \rfloor .

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