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1 < a_n < 1 +1/3^n if a_n =(2a_{n-1} + 1)/(a_{n-1} + 2)

Source: 2017 Grand Duchy of Lithuania, Mathematical Contest p1 (Baltic Way TST)

October 3, 2020
Sequencerecurrence relationinequalitiesalgebra

Problem Statement

The infinite sequence a0,a1,a2,a3,...a_0, a_1, a_2, a_3,... is defined by a0=2a_0 = 2 and an=2an−1+1an−1+2a_n =\frac{2a_{n-1} + 1}{a_{n-1} + 2} , n=1,2,3,...n = 1, 2, 3, ... Prove that 1<an<1+13n1 < a_n < 1 + \frac{1}{3^n} for all n=1,2,3,..n = 1, 2, 3, . .
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