MathDB
Sequence

Source: Baltic Way 2017 Problem 1

November 14, 2017
algebraInequalitySequenceConvexityeasy sequenceinduction

Problem Statement

Let a0,a1,a2,...a_0,a_1,a_2,... be an infinite sequence of real numbers satisfying an−1+an+12≥an\frac{a_{n-1}+a_{n+1}}{2}\geq a_n for all positive integers nn. Show that a0+an+12≥a1+a2+...+ann\frac{a_0+a_{n+1}}{2}\geq \frac{a_1+a_2+...+a_n}{n} holds for all positive integers nn.
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