Question

cho 1/a + 1/b + 1/c = 2 và 1/a^2 + 1+b^2 + 1/c^2 =2

CMR: a + b + c = abc

 

Answer 1

Ta có:

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=2\)

=> \(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2=2^2\)

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

=> \(2+2.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)=4\)

=> \(2.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)=2\)

=> \(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}=1\)

=> \(abc.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)=abc\)

=> \(c+a+b=abc\) (đpcm)

Answer 2

\(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2}{ab}+\frac{2}{ac}\)

\(=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\)

\(\Rightarrow2^2=2+2.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\)

\(\Leftrightarrow2=2.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\)

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

\(\Leftrightarrow a+b+c=abc\)

đpcm

\(\frac{\Leftrightarrow c}{abc}+\frac{a}{abc}+\frac{b}{abc}=\frac{abc}{abc}\)

Answer 3

Ta có:

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

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

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

=>2*\(\frac{a+b+c}{abc}\)=2

=>\(\frac{a+b+c}{abc}\)=1

=>a+b+c=abc