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a_n=\frac{1}{a_{n-1}-2}, b_n=\frac{2b_{n-1}+1} {b_{n-1}}

Source: China Northern MO 2012 p5 CNMO

May 4, 2024
Sequencerecurrence relationalgebra

Problem Statement

Let {an}\{a_n\} be the sequance with a0=0a_0=0, an=1an−1−2a_n=\frac{1}{a_{n-1}-2} (n∈N+n\in N_+). Select an arbitrary term aka_k in the sequence {an}\{a_n\} and construct the sequence {bn}\{b_n\}: b0=akb_0=a_k, bn=2bn−1+1bn−1b_n=\frac{2b_{n-1}+1} {b_{n-1}} (n∈N+n\in N_+) . Determine whether the sequence {bn}\{b_n\} is a finite sequence or an infinite sequence and give proof.
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