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a_{n+1} = (p_n^2+2015)(p_nq_n), bounded sequence?

Source: China Northern MO 2015 grade 11 p8 CNMO

May 5, 2024
Sequencealgebrabounded

Problem Statement

The sequence {an}\{a_n\} is defined as follows: a1a_1 is a positive rational number, an=pnqna_n= \frac{p_n}{q_n}, (n=1,2,…n= 1,2,…) is a positive integer, where pnp_n and qnq_n are positive integers that are relatively prime, then an+1=pn2+2015pnqna_{n+1} = \frac{p_n^2+2015}{p_nq_n} Is there a1>2015_1>2015, making the sequence {an}\{a_n\} a bounded sequence? Justify your conclusion.
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