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Show that every term of the sequence exceeds 1

Source: Canada National Mathematical Olympiad 1977 - Problem 6

September 29, 2011

quadraticsalgebraalgebra proposed

Problem Statement

Let

0 < u < 1

and define u_1 = 1 + u, u_2 = \frac{1}{u_1} + u, \dots, u_{n + 1} = \frac{1}{u_n} + u, n \ge 1. Show that

u_n > 1

for all values of

n = 1

, 2, 3,

\dots