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Finding harmonic progression in (1,4,7,10,...)

Source: RMO 2016 Hyderabad , P6 .

October 12, 2016

number theory

Problem Statement

Show that the infinite arithmetic progression

\{1,4,7,10 \ldots\}

has infinitely many 3 -term sub sequences in harmonic progression such that for any two such triples

\{a_1, a_2 , a_3 \}

and

\{b_1, b_2 ,b_3\}

in harmonic progression , one has

\frac{a_1} {b_1} \ne \frac {a_2}{b_2}