Can you prove this inequality? | USSR Olympiad Problem | Prove 𝒏! ≤ ((𝒏+𝟏)/𝟐)^𝒏

Prove 𝒏! ≤ ((𝒏+𝟏)/𝟐)^𝒏. We apply the Theorem of the Arithmetic and Geometric Means to complete the proof. Don't use Induction method.