MathDB

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

Source: Rioplatense Olympiad 2001 level 3 P3

September 6, 2018

algebrafloor functionfunctionSequenceradical

Problem Statement

For every integer

n > 1

, the sequence

\left( {{S}_{n}} \right)

is defined by

{{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor

where

\left\lfloor x \right\rfloor

denotes the floor function of

x

. Prove that

{{S}_{2001}}=2\,{{S}_{2000}}+1

. .