[Toán 8] Chứng minh đẳng thức

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vitconcatinh_foreverloveyou

[tex]\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c}[/tex]

[tex] \Leftrightarrow \frac{ab+bc+ca}{abc} = \frac{1}{a+b+c}[/tex]

[tex] \Leftrightarrow a^2b + ab^2 + abc + + bca + b^2c + bc^2 + ca^2 + cab + c^2 = abc[/tex]

[tex] \Leftrightarrow (a^2b + b^a) + (abc + b^2c) + (bca + ca^2) + (bc^2 + c^2a) = 0[/tex]

[tex] \Leftrightarrow ab(a+b) + bc(a+b) + ca(a+b) + c^2(a+b) = 0[/tex]

[tex] \Leftrightarrow (a+b)(ab+bc+ca+c^2) =0[/tex]

[tex] \Leftrightarrow (a+b)(b+c)(c+a) = 0[/tex]

[tex] \Leftrightarrow \left[\begin{a=-b}\\{b = -c}\\{c=-a} [/tex]

thay a = -b vào bt trên ta có

[tex] \frac{1}{a^7} + \frac{1}{b^7} + \frac{1}{c^7} = \frac{-1}{b^7} + \frac{1}{b^7} + \frac{1}{c^7} = \frac{1}{c^7}[/tex]

[tex] \frac{1}{a^7 + b^7 + c^7} = \frac{1}{c^7}[/tex]

cmtt với b=-c, c=-a

[tex] \rightarrow dpcm[/tex]
 
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