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for some k > 1: x_{k-1}y_{k-1} = 42, x_ky_k = 30, and x_{k+1}y_{k+1} = 16

Source: Czech and Slovak MO, III A, 2005 p1

January 12, 2020

arithmetic sequenceSequencealgebramaximumrecurrence relation

Problem Statement

Consider all arithmetical sequences of real numbers

(x_i)^{\infty}=1

and

(y_i)^{\infty} =1

with the common first term, such that for some

k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30

, and

x_{k+1}y_{k+1} = 16

. Find all such pairs of sequences with the maximum possible

k