<s>[INST] Show that $\frac{1}{a^n+b^n+c^n} = \frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}$ Let $n \in N, n=2k+1, and \text{ } \frac{1}{a+b+c} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$.

Show that $$\frac{1}{a^n+b^n+c^n} = \frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}$$

I have tried, but I don't get anything. Can you please give me a hint?
 [/INST] This method of infinite ascent is probably wrong.. but...
Suppose none of $a = -b, b = -c, c = -a$ is true. 
Then multiplying both sides by $abc$ and inverting we get:
$\frac{a+b+c}{abc} = \frac{1}{ab + bc + ca} \implies \frac{1}{ab + bc + ca}=  \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}$
We can now write $x = ab, y = bc, z = ca$ and do this all over again which is probably absurd (needs better logic here). 
</s>