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MMO 257 Moscow MO 1953 x_n=(x^2_{n-1}+2)/(2x_(n-1))

Source:

August 9, 2019

recurrence relationSequenceinequalitiesalgebra

Problem Statement

Let

x_0 = 10^9

x_n = \frac{x^2_{n-1}+2}{2x_{n-1}}

for

n > 0

. Prove that

0 < x_{36} - \sqrt2 < 10^{-9}