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\(M=\dfrac{10^{20}+1}{10^{19}+1}\)
\(N=\dfrac{10^{21}+1}{10^{20}+1}< \dfrac{10^{21}+1+9}{10^{20}+1+9}=\dfrac{10^{21}+10}{10^{20}+10}=\dfrac{10\left(10^{20}+1\right)}{10\left(10^{19}+1\right)}=\dfrac{10^{20}+1}{10^{19}+1}=M\)
\(\Rightarrow N< M\)
M = \(\dfrac{10^{20}+1}{10^{19}+1}\) = 10 - \(\dfrac{9}{10^{19}+1}\) ; N = \(\dfrac{10^{21}+1}{10^{20}+1}\) = 10 - \(\dfrac{9}{10^{20}+1}\)
Vì \(\dfrac{9}{10^{19}+1}\) > \(\dfrac{9}{10^{20}+1}\)
⇒ M < N (phân số nào có phần bù lớn hơn thì phân số đó nhỏ hơn)
\(M=\dfrac{10^{20}+1}{10^{19}+1}\)
\(N=\dfrac{10^{21}+1}{10^{20}+1}< \dfrac{10^{21}+1+9}{10^{20}+1+9}=\dfrac{10^{21}+10}{10^{20}+10}=\dfrac{10\left(10^{20}+1\right)}{10\left(10^{19}+1\right)}=\dfrac{10^{20}+1}{10^{19}+1}=M\)
\(\Rightarrow N< M\)
Ta dùng bất đẳng thức\(\frac{a}{b}<\frac{a+n}{b+n}\left(n\ne0\right)\)
Ta có \(B=\frac{10^{20}+1}{10^{21}+1}<\frac{10^{20}+1+9}{10^{21}+1+9}<\frac{10^{20}+10}{10^{21}+10}<\frac{10\left(10^{19}+1\right)}{10\left(10^{20}+1\right)}\)
\(<\frac{10^{19}+1}{10^{20}+1}\)
Vậy \(A>B\)
Ta có:\(B=\frac{10^{20}+1}{10^{21}+1}< 1\Rightarrow B=\frac{10^{20}+1}{10^{21}+1}< \frac{10^{20}+1+9}{10^{21}+1+9}=\frac{10^{20}+10}{10^{21}+10}=\frac{10\left(10^{19}+1\right)}{10\left(10^{20}+1\right)}=\frac{10^{19}+1}{10^{20}+1}=A\)
=> A > B
Ta có:
\(B=\frac{20^{10}-1}{20^{10}-3}>1\)
=> Ta có: \(B=\frac{20^{10}-1}{20^{10}-3}>\frac{20^{10}-1-2}{20^{10}-3-2}=\frac{20^{11}+1}{20^{10}+1}=A\)
\(\Rightarrow A< B\)
\(A=\frac{20^{10}+1}{20^{10}-1}=\frac{20^{10}-1+2}{20^{10}-1}=1+\frac{2}{20^{10}-1}\)
\(B=\frac{20^{10}-1}{20^{10}-3}=\frac{20^{10}-3+2}{20^{10}-3}=1+\frac{2}{20^{10}-3}\)
vì \(\frac{2}{20^{10}-1}\) < \(\frac{2}{20^{10}-3}\)
=> 1 + \(\frac{2}{20^{10}-1}\) < 1+\(\frac{2}{20^{10}-3}\)
hay A< B
Ta có :
\(\frac{10^{20}+1}{10^{21}+1}< \frac{10^{20}+1+9}{10^{21}+1+9}=\frac{10^{20}+10}{10^{21}+10}=\frac{10\left(10^{19}+1\right)}{10\left(10^{20}+1\right)}=\frac{10^{19}+1}{10^{20}+1}\)
Vậy \(\frac{10^{19}+1}{10^{20}+1}>\frac{10^{20}+1}{10^{21}+1}\)
Ta có: A=\(\frac{10^{20}+1}{10^{21}+1}\)< 1 => \(\frac{10^{20}+1+9}{10^{21}+1+9}\)<1 => \(\frac{10^{20}+1+9}{10^{21}+1+9}\) = \(\frac{10^{20}+10}{10^{21}+10}\)=\(\frac{10.\left(10^{19}+1\right)}{10.\left(10^{20}+1\right)}\)=\(\frac{10^{19}+1}{10^{20}+1}\)=>B<A
A=B =1/10