Question

Cho dãy tỉ số bằng nhau \(\frac{\text{ab}}{b}=\frac{bc}{c}=\frac{ca}{a}\). Chứng minh rằng \(a=b=c\)

Answer 1

ĐKXĐ :  a;b;c \(\ne0\)

Khi đó \(\frac{ab}{b}=\frac{bc}{c}=\frac{ca}{a}\)

<=> \(a.\frac{b}{b}=b.\frac{c}{c}=c.\frac{a}{a}\)

<=> \(a=b=c\)

Answer 2

Từ: \(\frac{ab}{b}=\frac{bc}{c}=\frac{ca}{a}\Leftrightarrow\frac{a}{b}=\frac{b}{c}=\frac{c}{a}\left(đk: a,b,c>0; a+b+c\ne0\right)\)

Có: \(\frac{a}{b}=\frac{b}{c}=\frac{c}{a}=\frac{a+b+c}{b+c+a}=1\left(a+b+c\ne0\right)\Leftrightarrow a=b=c\)

Answer 3

Có: \(\frac{ab}{b}=a\)

\(\frac{bc}{c}=c\)

\(\frac{ca}{a}=c\)

Do đó, a=b=c (sai thông cảm)

Answer 4

\(\frac{\overline{ab}}{b}=\frac{\overline{bc}}{c}+\frac{\overline{ca}}{a}=\frac{10a+b}{b}=\frac{10b+c}{c}=\frac{10c+a}{a}=\)

\(=\frac{10a+b+10b+c+10c+a}{a+b+c}=11\)

\(\Rightarrow\frac{10a+b}{b}=11\Rightarrow10a+b=11b\Rightarrow10a=10b\Rightarrow a=b\)

\(\Rightarrow\frac{10b+c}{c}=11\Rightarrow10b+c=11c\Rightarrow10b=10c\Rightarrow b=c\)

\(\Rightarrow a=b=c\)