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1 -1/2^{2^{n-1}}<sum 1/x+i<1/2^{2^{n}} if x_{n+1} = x_n^2 - x_n + 1, x_1 = 2

Source: 2018 Saudi Arabia GMO TST I p1

July 31, 2020

Sequencealgebrainequalitiesrecurrence relationreciprocal

Problem Statement

Let

\{x_n\}

be a sequence defined by

x_1 = 2

and

x_{n+1} = x_n^2 - x_n + 1

for

n \ge 1

. Prove that

1 -\frac{1}{2^{2^{n-1}}} < \frac{1}{x_1}+\frac{1}{x_2}+ ... +\frac{1}{x_n}< 1 -\frac{1}{2^{2^n}}

for all

n