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Từ 1/a + 1/b + 1/c = 2 bình phương hai vế ta có:
(1/a + 1/b + 1/c)² = 2²
=> 1/a² + 1/b² + 1/c² + 2(1/ab + 1/bc + 1/ ca) = 4
=> 1/a² + 1/b² + 1/c² + 2(a + b + c)/abc = 4 (Quy đồng MTC= abc)
=> 1/a² + 1/b² + 1/c² + 2abc/abc = 4 (Vì a + b + c = abc)
=> 1/a² + 1/b² + 1/c² + 2 = 4
=> 1/a² + 1/b² + 1/c² = 2
Vậy, P= 2
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ta có: a+b+c = abc
\(\Rightarrow\frac{a+b+c}{abc}=1\)
\(\Rightarrow\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}=1\)
Lại có: \(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\)
\(2^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2.1\)
\(\Rightarrow\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=2\)
Cho a+b+c=abc và 1/a+1/b+1/c=2.
CMR: 1/a^2 +1/b^2 +1/c^2 =2
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\(abc=a+b+c\Leftrightarrow\frac{abc}{abc}=\frac{a+b+c}{abc}\)
\(\Leftrightarrow1=\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}=Q\)
\(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)\)
\(\Rightarrow P=3^2-2Q=9-2=7\)
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1) ta có: a(b^2 -1)(c^2 -1)+b(a^2 -1)(c^2 -1)+c(a^2-1)(b^2-1)
=(ab^2 -a)(c^2-1)+(ba^2 -b)(c^2-1)+(ca^2-c)(b^2-1)
đén đây nhân bung ra hết rồi rút gọn và thay a+b+c=abc là đc
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Ta có A=\(\left(ab+bc+ca\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-abc\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\)
=\(2\left(a+b+c\right)+\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}-\frac{ab}{c}-\frac{bc}{a}-\frac{ca}{b}=2\left(a+b+c\right)\)
\(A=\left(a+b\right)\left(a^2-ab+b^2\right)+3ab\left[\left(a+b\right)^2-2ab\right]+6a^2b^2=a^2-ab+b^2+3ab\left(1-2ab\right)+6a^2b^2\)
=\(\left(a+b\right)^2-3ab+3ab-6a^2b^2+6a^2b^2=1\)
2) Ta có \(A=\left(a-1\right)\left(b-1\right)\left(c-1\right)=abc-ab-bc-ca+a+b+c-1=0\)
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)
\(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2}{ab}+\frac{2}{ac}\)
\(=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\)
\(\Rightarrow2^2=2+2.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\)
\(\Leftrightarrow2=2.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\)
\(\Leftrightarrow\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}=1\)
\(\Leftrightarrow a+b+c=abc\)
đpcm
\(\frac{\Leftrightarrow c}{abc}+\frac{a}{abc}+\frac{b}{abc}=\frac{abc}{abc}\)