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มาเติมโจทย์ครับ

51. Let $a,b,c$ is nonzero real numbers such that $a+b+c\not=0$ and $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$$
Prove that for all odd integers $n$ $$\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$$