Prove that

Question

x₁/x₂ + x₂/x₃ + .......... + x₂₀₁/x₁ ≥ 201

Answer 1

I'm assuming the x_i are all positive. By the arithmetic-geometric mean inequality, the expression on the left hand side of your inequality is greater than or equal to 

 

201[(x₁/x₂)(x₂/x₃)...(x₂₀₀/x₂₀₁)(x₂₀₁/x₁)]^1/201 = 201[(x₁x₂...x₂₀₁)/(x₁x₂...x₂₀₁)]^1/201 = 201.