Question

Cho dãy tỉ số bằng nhau \(\frac{\overline{ab}}{b}=\frac{\overline{bc}}{c}=\frac{\overline{ca}}{a}\).CMR::\(a=b=c\)

Answer 1

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

áp dụng t/c dãy tỉ số bằng nhau ta có:

\(\frac{\overline{ab}}{b}=\frac{\overline{bc}}{c}=\frac{\overline{ca}}{a}=\frac{10a+b}{b}=\frac{10b+c}{c}=\frac{10c+a}{a}=\frac{11.\left(a+b+c\right)}{a+b+c}=11\)

\(\frac{10a+b}{b}=11\Rightarrow10a+b=11b\Rightarrow10a=10b\Rightarrow a=b\)(1)

\(\frac{10b+c}{c}=11\Rightarrow10b+c=11c\Rightarrow10b=10c\Rightarrow b=c\)(2)

\(\frac{10c+a}{a}=11\Rightarrow10c+a=11a\Rightarrow10c=10a\Rightarrow c=a\)(3)

từ (1), (2), (3)  => a=b=c (đpcm)