Radically Compromised

Geometry Level 5

an+1=2+an\large a_{n+1} = \sqrt{2+a_n}

Consider the recurrence relation ana_n above for n=1,2,3,n = 1,2, 3, \ldots and a1=3a_1 = \sqrt3.

Evaluate log2[cos1(a20162)] \log_2 \left [ \cos^{-1} \left ( \dfrac{a_{2016}}2 \right) \right ] .

If this number can be expressed as log2πlog2ab \log_2 \pi - \log_2 a - b, where aa and bb are positive integers with aa odd, submit your answer as a+ba+b.

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