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Prove that it is a geometric sequence

Source: Romanian MO 2010 Grade 10

August 6, 2012
quadraticsVietafunctionalgebraalgebra unsolved

Problem Statement

Let (an)n≥0(a_n)_{n\ge0} be a sequence of positive real numbers such that ∑k=0nCnkakan−k=an2, for any n≥0.\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0. Prove that (an)n≥0(a_n)_{n\ge0} is a geometric sequence.
Lucian Dragomir
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