Tính b, c biết \(\frac{a}{b} = \frac{b}{c} = \frac{c}{a},\;a = 2005\) và a + b + c ¹ 0.

Lời giải

ĐK: b, c ¹ 0

Do \(\frac{a}{b} = \frac{b}{c} = \frac{c}{a},\;a = 2005\) nên suy ra \(\frac{{2005}}{b} = \frac{b}{c} = \frac{c}{{2005}}\)

\[ \Rightarrow \left\{ \begin{array}{l}{b^2} = 2005c\\{c^2} = 2005b\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}{b^2} = 2005c\\{\left( {\frac{b}{c}} \right)^2} = \frac{c}{b}\end{array} \right.\]

\( \Leftrightarrow \left\{ \begin{array}{l}{b^2} = 2005c\\{\left( {\frac{b}{c}} \right)^3} = 1\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}{b^2} = 2005c\\\frac{b}{c} = 1\end{array} \right.\)

\[ \Leftrightarrow \left\{ \begin{array}{l}{b^2} = 2005b\\b = c\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}b = c = 2005\;\;\;\left( {TM} \right)\\b = c = 0\;\;\;\;\;\;\;\;\;\left( l \right)\end{array} \right.\]

Vậy b = c = 2005