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\(A=\frac{10^{2012}+1}{10^{2013}+1}\)
\(10A=\frac{10\cdot\left[10^{2012}+1\right]}{10^{2013}+1}=\frac{10^{2013}+10}{10^{2013}+1}=\frac{10^{2013}+1+9}{10^{2013}+1}=1+\frac{9}{10^{2013}+1}\)
\(B=\frac{10^{2013}+1}{10^{2014}+1}\)
\(10B=\frac{10\cdot\left[10^{2013}+1\right]}{10^{2014}+1}=\frac{10^{2014}+10}{10^{2014}+1}=\frac{10^{2014}+1+9}{10^{2014}+1}=1+\frac{9}{10^{2014}+1}\)
Mà \(1+\frac{9}{10^{2013}+1}>1+\frac{9}{10^{2014}+1}\)
Nên \(10A>10B\)
Hay \(A>B\)
Vậy : A > B
vì B<1 => \(B=\frac{10^{2013}+1}{10^{2014}+1}< \frac{10^{2013}+1+9}{10^{2014}+1+9}=\)\(\frac{10^{2013}+10}{10^{2014}+10}=\frac{10\left(10^{2012}+1\right)}{10\left(10^{2013}+1\right)}\)\(=\frac{10^{2012}+1}{10^{2013}+1}=A\)
\(\Rightarrow A>B\)
\(\frac{10^{2012}+1}{10^{2013}+1}=\frac{\left(10^{2012}+1\right)\cdot10}{\left(10^{2013}+1\right)\cdot10}=\frac{10^{2013}+10}{\left(10^{2013}+1\right)\cdot10}=\frac{10^{2013}+1+9}{\left(10^{2013}+1\right)\cdot10}=\frac{10^{2013}+1}{\left(10^{2013}+1\right)\cdot10}+\frac{9}{\left(10^{2013}+1\right)\cdot10}=\frac{1}{10}+\frac{9}{\left(10^{2013}+1\right)\cdot10}\left(1\right)\)
\(\frac{10^{2013}+1}{10^{2014}+1}=\frac{\left(10^{2013}+1\right)\cdot10}{\left(10^{2014}+1\right)\cdot10}=\frac{10^{2014}+10}{\left(10^{2014}+1\right)\cdot10}=\frac{10^{2014}+1+9}{\left(10^{2014}+1\right)\cdot10}=\frac{10^{2014}+1}{\left(10^{2014}+1\right)\cdot10}+\frac{9}{\left(10^{2014}+1\right)\cdot10}=\frac{1}{10}+\frac{9}{\left(10^{2014}+1\right)\cdot10}\left(2\right)\)Từ (1)(2) => A > B
\(\Rightarrow10A=10.\left(\frac{10^{2012}+1}{10^{2013}+1}\right)=\frac{10^{2013}+10}{10^{2013}+1}=\frac{10^{2013}+1+9}{10^{2013}+1}=1+\frac{9}{10^{2013}+1}\)
\(\Rightarrow10B=10.\left(\frac{10^{2013}+1}{10^{2014}+1}\right)=\frac{10^{2014}+10}{10^{2014}+1}=\frac{10^{2014}+1+9}{10^{2014}+1}=1+\frac{9}{10^{2014}+1}\)
Ta có: 1 = 1; 9 = 9
Mà \(10^{2013}+1<10^{2014}+1\)
=> \(\frac{9}{10^{2013}+1}>\frac{9}{10^{2014}+1}\)
=> \(1+\frac{9}{10^{2013}+1}>1+\frac{9}{10^{2014}+1}\text{ hay }10A>10B\)
=> \(A>B\).
a) \(\frac{2^{10}+1}{2^{10}-1}\)và \(\frac{2^{10}-1}{2^{10}-3}\)
Ta có chính chất phân số trung gian là \(\frac{2^{10}+1}{2^{10}-3}\)
\(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}+1}{2^{10}-3}\) ; \(\frac{2^{10}-1}{2^{10}-3}< \frac{2^{10}+1}{2^{10}-3}\)
Vì \(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}+1}{2^{10}-3}>\frac{2^{10}-1}{2^{10}-3}\)
Nên \(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}-1}{2^{10}-3}\)
b) \(A=\frac{2011}{2012}+\frac{2012}{2013}\)và \(B=\frac{2011+2012}{2012+2013}\)
Ta có : \(A=\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{2013}+\frac{2012}{2013}=\frac{2011+2012}{2013}>\frac{2011+2012}{2012+2013}=B\)
Vậy A > B
Có gì sai cho sorry
a,
\(\frac{2^{10}+1}{2^{10}-1}=1+\frac{2}{2^{10}-1}< 1+\frac{2}{2^{10}-3}=\frac{2^{10}-1}{2^{10}-3}\)
b,
\(\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{2012+2013}+\frac{2012}{2012+2013}=\frac{2011+2012}{2012+2013}\)
2. TA CÓ: D=\(\frac{2011+2012}{2012+2013}\)
=\(\frac{2011}{2012+2013}+\frac{2012}{2012+2013}\)
VÌ 2012+2013>2012
MÀ \(\frac{2011}{2012+2013}
Ta có công thức :
\(\frac{a}{b}< \frac{a+c}{b+c}\)\(\left(\frac{a}{b}< 1;a,b,c\inℕ^∗\right)\)
Áp dụng vào ta có :
\(B=\frac{10^{2014}+1}{10^{2015}+1}< \frac{10^{2014}+1+9}{10^{2015}+1+9}=\frac{10^{2014}+10}{10^{2015}+10}=\frac{10\left(10^{2013}+1\right)}{10\left(10^{2014}+1\right)}=\frac{10^{2013}+1}{10^{2014}+1}=A\)
\(\Rightarrow\)\(B< A\) hay \(A>B\)
Vậy \(A>B\)
Chúc bạn học tốt ~
áp dụng tính chất
nếu a/b>1thì a/b<(a+n)/(b+n)
=)))))))))))))))))
TA có :
A = \(\frac{10^{2012}-2}{10^{2013}-1}\)=> 10A = \(1-\frac{19}{10^{2013}-1}\)
B = \(\frac{10^{2013}-2}{10^{2014}-1}\)=> 10B = 1 - \(\frac{19}{10^{2014}-1}\)
Vì \(1-\frac{19}{10^{2013}-1}\)< 1 - \(\frac{19}{10^{2014}-1}\)hay 10A < 10B => A < B
Vậy A < B