Question

Cho A =\(\frac{10^{11}-1}{10^{12}-1}\);      B =\(\frac{10^{12}-1}{10^{13}-1}\) . So sánh A và B.

Answer 1

\(10A=\frac{10\left(10^{11}-1\right)}{10^{12}-1}=\frac{10^{12}-10}{10^{12}-1}=\frac{10^{12}-1-9}{10^{12}-1}=1-\frac{9}{10^{12}-1}\)

\(10B=\frac{10\left(10^{12}-1\right)}{10^{13}-1}=\frac{10^{13}-10}{10^{13}-1}=\frac{10^{13}-1-9}{10^{13}-1}=1-\frac{9}{10^{13}-1}\)

Vì \(10^{13}-1>10^{12}-1\Rightarrow\frac{9}{10^{13}-1}< \frac{9}{10^{12}-1}\Rightarrow-\frac{9}{10^{13}-1}>-\frac{9}{10^{12}-1}\)

\(\Rightarrow1-\frac{9}{10^{13}-1}>1-\frac{9}{10^{12}-1}\Rightarrow10B>10A\Rightarrow B>A\)

Answer 2

\(A=\frac{10^{11}-1}{10^{12}-1}\Leftrightarrow10A=\frac{10^{12}-10}{10^{12}-1}=\frac{10^{12}-1-9}{10^{12}-1}=1-\frac{9}{10^{12}-1}\)

\(B=\frac{10^{12}-1}{10^{13}-1}\Leftrightarrow10B=\frac{10^{13}-10}{10^{13}-1}=\frac{10^{13}-1-9}{10^{13}-1}=1-\frac{9}{10^{13}-1}\)

\(\text{Vì }1-\frac{9}{10^{12}-1}< 1-\frac{9}{10^{13}-1}\Rightarrow10A< 10B\)

\(\Rightarrow A< B\)

Answer 3

Đầu tiên,ta c/m bđt phụ: Với \(\frac{a}{b}< 1\Rightarrow\frac{a}{b}< \frac{a+m}{b+m}\left(m>0\right)\)

Thật vậy: \(\frac{a}{b}< 1\Leftrightarrow a< b\)

Điều cần chứng minh tương đương với: \(a\left(b+m\right)< b\left(a+m\right)\)

\(\Leftrightarrow ab+am< ab+bm\)

\(\Leftrightarrow am< bm\Leftrightarrow a< b\) (đúng)

Vậy \(\frac{a}{b}< 1\Rightarrow\frac{a}{b}< \frac{a+m}{b+m}\left(m>0\right)\)

\(A=\frac{10^{11}-1}{10^{12}-1}=\frac{10\left(10^{11}-1\right)}{10\left(10^{12}-1\right)}=\frac{10^{12}-10}{10^{13}-10}\)

Mà \(\frac{10^{12}-10}{10^{13}-10}< 1\) nên \(\frac{10^{12}-10}{10^{13}-10}< \frac{10^{12}-10+9}{10^{13}-10+9}=\frac{10^{12}-1}{10^{13}-1}=B\)