MathDB

sequence positive

Source: ISL 2006, A2, VAIMO 2007, P4, Poland 2007

April 22, 2007

integrationinequalitiesalgebraSequenceSummationcalculusIMO Shortlist

Problem Statement

The sequence of real numbers

a_0,a_1,a_2,\ldots

is defined recursively by a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0 \text{for} n\geq 1.Show that

a_{n} > 0

for all

n\geq 1

.
Proposed by Mariusz Skalba, Poland