Question

cho a,b,c khác 0 Biết:1/a+1/b+1/c=2 1/a^2+1/b^2+1/c^2=2; chứng minh a+b+c=abc

Answer 1

Lời giải:

$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=2$

$\Rightarrow (\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^2=4$

$\Leftrightarrow \frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac})=4$

$\Leftrightarrow 2+2(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac})=4$

$\Leftrightarrow \frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}=1$

$\Leftrightarrow \frac{a+b+c}{abc}=1$

$\Leftrightarrow a+b+c=abc$ (đpcm)