# State and prove a natural generalization

1. Sep 22, 2011

### major_maths

1. The problem statement, all variables and given/known data
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.

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

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?

2. Sep 22, 2011

### 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?

3. Sep 22, 2011

### major_maths

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

Last edited: Sep 22, 2011
4. Sep 23, 2011

### spamiam

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.