Prove Inequality: a,b,c ∈R+ | n≥1

[Kryna]

Homework Statement

Prove $$\frac{a^{n+1}}{b+c}+\frac{b^{n+1}}{a+c}+\frac{c^{n+1}}{a+b}=(\frac{a^{n}}{b+c}+\frac{b^{n}}{a+c}+\frac{c^{n}}{a+b})*\sqrt[n]{\frac{a^{n}+b^{n}+c^{n}}{3}}$$
if n>=1 and a,b,c $$\in\textsl{R}_{+}$$

Homework Equations

The Attempt at a Solution

I tried prove it i some ways but i think any of it don't approach me to solution. I need a clue, don't give me solution.

PS sorry for my english

 

[Mark44]

Clue: mathematical induction

 

[Kryna]

Mathematical Induction is a method of proving a series of mathematical statement labelled by natural numbers

 

[Mark44]

Right, and aren't your values of n natural numbers? There is no such restriction on a, b, and c.

 

[Kryna]

I get

$$a^{2}+b^{2}+c^{2}\geq ab+bc+ac$$ for n=1
is it true?
what is next step(i never used mathematical induction before)

if i do it for n=2 it will be proved?

 

[Mark44]

For n = 1 you have to show that
$$\frac{a^{2}}{b+c}+\frac{b^{2}}{a+c}+\frac{c^{2}}{a+b}=(\frac{a^{1}}{b+c}+\frac{b^{1}}{a+c}+\frac{c^{1}}{a+b})*\frac{a^{1}+b^{1}+c^{1}}{3}$$

It is not sufficient to quit after showing that the original statement is true for n = 2.

In mathematical induction, you assume that the statement is true for n = k, and use that to show that the statement is also true for n = k + 1.

 

[Kryna]

Thanks for helping me.