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Inequality with a sequence

Source: 2019 Belarus Team Selection Test 1.4

September 2, 2019
inequalitiesalgebraSequences

Problem Statement

Let the sequence (an)(a_n) be constructed in the following way: a_1=1,\mbox{ }a_2=1,\mbox{ }a_{n+2}=a_{n+1}+\frac{1}{a_n},\mbox{ }n=1,2,\ldots. Prove that a180>19a_{180}>19.
(Folklore)
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