An Arithmetic Product Bound

★ 8/10Arithmetic Progression sequences AM-GM inequality Symmetry

Problem

Let $\{a_n\}$ be an arithmetic sequence with all terms positive. Prove that

\left(1 + \frac{1}{a_1}\right)\left(1 + \frac{1}{a_2}\right)\cdots\left(1 + \frac{1}{a_n}\right) \leqslant \left(1 + \frac{a_1 + a_n}{2a_1a_n}\right)^n.

Answer

Solution

Difficulty	8/10
Topics	Arithmetic Progression, sequences, AM-GM, inequality, Symmetry

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