Giải thích các bước giải:
$DK:{x_1},{x_2},{x_3},{x_4},{x_5},{x_6} \ne 0$
Ta có:
$\begin{array}{l}
x_2^2 = {x_1}{x_3} \Rightarrow \dfrac{{{x_1}}}{{{x_2}}} = \dfrac{{{x_2}}}{{{x_3}}}\\
x_3^2 = {x_2}{x_4} \Rightarrow \dfrac{{{x_2}}}{{{x_3}}} = \dfrac{{{x_3}}}{{{x_4}}}\\
x_4^2 = {x_3}{x_5} \Rightarrow \dfrac{{{x_3}}}{{{x_4}}} = \dfrac{{{x_4}}}{{{x_5}}}\\
x_5^2 = {x_4}{x_6} \Rightarrow \dfrac{{{x_4}}}{{{x_5}}} = \dfrac{{{x_5}}}{{{x_6}}}
\end{array}$
Như vậy ta có:
$\dfrac{{{x_1}}}{{{x_2}}} = \dfrac{{{x_2}}}{{{x_3}}} = \dfrac{{{x_3}}}{{{x_4}}} = \dfrac{{{x_4}}}{{{x_5}}} = \dfrac{{{x_5}}}{{{x_6}}}$
$ \Rightarrow \dfrac{{{x_1}}}{{{x_2}}} = \dfrac{{{x_2}}}{{{x_3}}} = \dfrac{{{x_3}}}{{{x_4}}} = \dfrac{{{x_4}}}{{{x_5}}} = \dfrac{{{x_5}}}{{{x_6}}} = \dfrac{{{x_1} + {x_2} + {x_3} + {x_4} + {x_5}}}{{{x_2} + {x_3} + {x_4} + {x_5} + {x_6}}}$
$ \Rightarrow \dfrac{{{x_1}}}{{{x_6}}} = \left( {\dfrac{{{x_1}}}{{{x_2}}}.\dfrac{{{x_2}}}{{{x_3}}}.\dfrac{{{x_3}}}{{{x_4}}}.\dfrac{{{x_4}}}{{{x_5}}}.\dfrac{{{x_5}}}{{{x_6}}}} \right) = {\left( {\dfrac{{{x_1}}}{{{x_2}}}} \right)^5} = {\left( {\dfrac{{{x_1} + {x_2} + {x_3} + {x_4} + {x_5}}}{{{x_2} + {x_3} + {x_4} + {x_5} + {x_6}}}} \right)^5}$
Ta có đpcm.
