Question

Ai giúp mình với

Cho 3 số a,b,c# 0 và (a+b+c)^2=a^2+b^2+c^2.Chứng minh rằng : 1/a^3+1/b^3+1/c^3=3/abc

thanks nha!!))

Answer 1

Từ \(\left(a+b+c\right)^2=a^2+b^2+c^2+2ab+2ac+2bc\)

Mà \(\left(a+b+c\right)^2=a^2+b^2+c^2\)

\(\Rightarrow2ab+2ac+2bc=0\)

\(\Rightarrow2\left(ab+ac+bc\right)=0\)

\(\Rightarrow ab+ac+bc=0\)

\(\Leftrightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\Leftrightarrow\frac{1}{a}=-\left(\frac{1}{b}+\frac{1}{c}\right)\). Khi đó

\(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=\frac{1}{b^3}+\frac{1}{c^3}-\left(\frac{1}{b}+\frac{1}{c}\right)^3=-\frac{3}{bc}\left(\frac{1}{b}+\frac{1}{c}\right)=-\frac{3}{bc}\cdot\frac{-1}{a}=\frac{3}{abc}\)