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sequence bounding

Source: Italy TST 2000

September 27, 2008

inductionalgebra unsolvedalgebra

Problem Statement

Given positive numbers

a_1

and

b_1

, consider the sequences defined by a_{n+1}=a_n+\frac{1}{b_n}, b_{n+1}=b_n+\frac{1}{a_n} (n \ge 1) Prove that

a_{25}+b_{25} \geq 10\sqrt{2}