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find the minimum

Source: 2022 China Southeast Grade 10 P5

August 3, 2022
inequalitiesNumber sequenceGCD

Problem Statement

Positive sequences {an},{bn}\{a_n\},\{b_n\} satisfy:a1=b1=1,bn=anbn−1−14(n≥2)a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2). Find the minimum value of 4b1b2⋯bm+∑k=1m1a1a2⋯ak4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k},where mm is a given positive integer.
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