A Log-Squeezed Sum

★ 8/10sequences Logarithms AM-GM Telescoping inequality

Problem

A positive sequence has $a_1 = 1$ and

a_{n+1} = \frac{a_n}{\sqrt{a_n^2 + 1}}.

Prove that

\ln(n+1) < a_1a_2 + a_2a_3 + \cdots + a_na_{n+1} < \ln\left(\frac{2n}{3} + 1\right) + \frac12.

Answer

Solution

Difficulty	8/10
Topics	sequences, Logarithms, AM-GM, Telescoping, inequality

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