State and prove a natural generalization

[major_maths]

Homework Statement

State and prove a natural generalization of "prove that for any three positive real numbers x₁, x₂, x₃, x₁/x₂ + x₂/x₃ + x₃/x₁ $\geq$ 3.

Homework Equations

AG Inequality is used in subproof (x₁/x₂ + x₂/x₃ + x₃/x₁ $\geq$ 3)

The Attempt at a Solution

I don't know what the book means exactly by a "natural generalization". Does it want me to prove the original AG Inequality or relate it somehow to this specific instance of the AG Inequality?

 

[spamiam]

What's special about the statement being true for 3 numbers? Is it true that for any 4 positive real numbers $x_1, x_2, x_3, x_4$
$$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_4}+ \frac{x_4}{x_1} \geq 4 \; ?$$
Can you generalize?

 

[major_maths]

So it's asking for a proof of the form (x₁+x₂+...+x_n)/n $\geq$ [itex]\sqrt[n]{x₁x₂...x_n}[/itex] . So, I should prove the Arithmetic-Geometric Mean Inequality?

 

[spamiam]

So it's asking for a proof of the form (x₁+x₂+...+x_n)/n $\geq$ [itex]\sqrt[n]{x₁x₂...x_n}[/itex] . So, I should prove the Arithmetic-Geometric Mean Inequality?

If by "of the form," you mean for $n$ positive real numbers, then yes. I'm guessing the same trick you used for the $n=3$ case will work in general. If not, proof by induction might work.