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Part of 2001 Rioplatense Mathematical Olympiad, Level 3

Problems(1)

another sequence with floor function, prove S_{2001}=2S_{2000}+1

Source: Rioplatense Olympiad 2001 level 3 P3

9/6/2018
For every integer n>1n > 1, the sequence (Sn)\left( {{S}_{n}} \right) is defined by Sn=⌊2n2+2+...+2⏟n radicals⌋{{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor where ⌊x⌋\left\lfloor x \right\rfloor denotes the floor function of xx. Prove that S2001=2 S2000+1{{S}_{2001}}=2\,{{S}_{2000}}+1. .
algebrafloor functionfunctionSequenceradical