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# Oh wow!

If $$\quad a^3+b^3+c^3 = (a+b+c)^3$$ ,

Prove that $$\quad \large a^{2n+1}+ b^{2n+1}+ c^{2n+1} = (a+b+c)^{2n+1}$$ for all $$n \in N$$

Nice proofs are always welcome. :)

Note by Nihar Mahajan
2 years, 3 months ago

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Well I THINK it can be proved without induction.@Nihar Mahajan @Vaibhav Prasad I will write it in short. It can easily be derived that $$(a+b)(b+c)(a+c)=0$$ Now we have 3 cases in first all the 3 terms are 0 , or any 2terms are 0 or only one term is 0.

Case$$(a+b)=0$$

$$a=-b$$

So now just substitute a = -b in the equation (which is to be proved)

That will give $$c^{2n+1} = c^{2n+1}$$ · 2 years, 3 months ago

Oh ,,, I got your method... Thanks! · 2 years, 3 months ago

See my approach:

$a^3+b^3+c^3=(a+b+c)^3-3(a+b+c)(ab+bc+ac)+3abc \\ \Rightarrow 3(a+b+c)(ab+bc+ac) = 3abc \\ (a+b+c)\left(\dfrac{ab+bc+ac}{abc}\right)=1 \quad or \quad (a+b+c)\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)=1$

Am not able to do more than this... · 2 years, 3 months ago

@Nihar Mahajan

Take $$a+c=x$$ Then $$(b+x)(bx+ac) = abc$$

Multiplying and simplifying gives us $$x(b^{2}+bx+ac )= 0$$substitute x Then Further factorizing gives us $$a+b)(b+c)(a+c) = 0$$

Now the method goes as usual (given in my method). · 2 years, 3 months ago

@Kalash Verma I understood your method. I am asking , can you extend my approach furthur? · 2 years, 3 months ago

@Nihar Mahajan Well actually its the extension of ur approach. In the third line I have replaced a+c by x and then I have explained above.In the end both results in same factors . · 2 years, 3 months ago

Thanks for your help! :) @Kalash Verma · 2 years, 3 months ago

Don't mention it.AND you have 703 followers and 703 following. Wow.congrats · 2 years, 3 months ago

I will try using this method and tell you about it . But this method does not work when a or b or c=0 and if its right then it proves 0/0=1 (lol was it so easy) · 2 years, 3 months ago

Yeah , my method has $$a,b,c \neq 0$$. · 2 years, 3 months ago

I will try solving it by your method and tell you tomorrow (well If I can solve it). · 2 years, 3 months ago

Well, if the first condition is met, then one of these are true, $$a+b=0$$, $$b+c=0$$, $$c+a=0$$, and so the next condition follows immediately. · 2 years, 3 months ago

@Michael Mendrin @Calvin Lin Sir , can you extend my approach that I have written above? · 2 years, 3 months ago

I can't go anywhere with that approach either. But if I just solve it for $$=0$$, I get the condition $$a+b=0$$, $$b+c=0$$, or $$c+a=0$$, and so the rest becomes easy. · 2 years, 3 months ago

Comment deleted Apr 20, 2015

Where has $$3abc$$ gone? · 2 years, 3 months ago

Induction on $$n$$ ?? · 2 years, 3 months ago

Most probably .. xD · 2 years, 3 months ago