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Phân số A lớn hơn đấy bạn ah .Mình chắc chắn đây,nhớ k cho mình nha
\(B=\frac{2011+2012}{2012+2013}=\frac{2011}{2012+2013}+\frac{2012}{2012+2013}\)
\(\frac{2011}{2012}>\frac{2011}{2012+2013};\frac{2012}{2013}>\frac{2012}{2012+2013}\)
Vậy A > B
a, \(\frac{2011}{2012}\)và \(\frac{2012}{2011}\)
Vì \(\frac{2011}{2012}\)có Tử số bé hơn Mẫu số nên phân số đó < 1 ; \(\frac{2012}{2011}\)có Tử số lớn hơn Mẫu số nên phân số đó > 1
=> \(\frac{2011}{2012}< \frac{2012}{2011}\)
b, \(\frac{2000}{2013}\)và \(\frac{2011}{2012}\)
Ta có:
\(\frac{2000}{2013}=\frac{2000}{2013}+\frac{13}{2013}\) ; \(\frac{2011}{2012}=\frac{2011}{2012}+\frac{1}{2012}\)
Ta thấy \(\frac{13}{2013}>\frac{1}{2012}\)
\(\Rightarrow\frac{2000}{2013}< \frac{2011}{2012}\)
Áp dụng BĐT \(\frac{a}{b}+\frac{b}{c}+\frac{c}{d}>\frac{a+b+c}{a+b+c}=1>\frac{a+b+c}{b+c+d}\).
\(\Rightarrow\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2013}>\frac{2010+2011+2012}{2010+2011+2012}>\frac{2010+2011+2012}{2011+2012+2013}\)mà 2010 + 2011 + 2012 < 2011+2012+2013 ,suy ra \(\frac{2010+2011+2012}{2011+2012+2013}< 1\))
\(\Rightarrow\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2013}>\frac{2010+2011+2012}{2011+2012+2013}\)hay P > Q
Vậy P > Q
b) Áp dụng công thức BCNN (a, b) . UCLN (a,b) = a.b
\(\Rightarrow a.b=420.21=8820\)
Ta có:
\(ab=8820\)
\(a+21=b\Rightarrow b-a=21\)
Hai số cách nhau 21 mà có tích là 8820 là 84 , 105
Mà a + 21 = b suy ra a < b
Vậy a = 84 ; b = 105
a,-Cách khác:
-Ta có: \(\frac{2010+2011+2012}{2011+2012+2013}=\frac{2010}{2011+2012+2013}+\frac{2011}{2011+2012+2013}+\frac{2012}{2011+2012+2013}\)
-Mà: \(\frac{2010}{2011}>\frac{2010}{2011+2012+2013}\left(1\right)\)
\(\frac{2011}{2012}>\frac{2011}{2011+2012+2013}\left(2\right)\)
\(\frac{2012}{2013}>\frac{2012}{2011+2012+2013}\left(3\right)\)
\(\Rightarrow P>Q\)
a. Ta có
\(B=\frac{2011+2012}{2012+2013}=\frac{2011}{2012+2013}+\frac{2012}{2012+2013}.\)
Vì\(\frac{2011}{2012+2013}< \frac{2011}{2012}.\)(1)
\(\frac{2012}{2012+2013}< \frac{2012}{2013}.\)(2)
Cộng vế với vế của 1;2 ta được
\(B=\frac{2011}{2012+2013}+\frac{2012}{2012+2013}< A=\frac{2011}{2012}+\frac{2012}{2013}\)
hay A>B
ÁP DỤNG CÔNG THỨC NẾU \(\frac{a}{b}\)>1 thì
\(\frac{a}{b}\)>\(\frac{a+m}{b+m}\)
Ta có : \(\frac{2012^{12}+1}{2012^{13}+1}\)>\(\frac{2012^{12}+1+2011}{2012^{13}+1+2011}\)=\(\frac{2012^{12}+2012}{2012^{13}+2012}\)=\(\frac{2012.\left(2012^{11}+1\right)}{2012.\left(2012^{12}+1\right)}\)
rồi rút gọn thành \(\frac{2012^{11}+1}{2012^{12}+1}=B\)
Vậy A>B
Nhớ cho mình đúng nha
Ta có:\(A=\dfrac{2012^{2012}+1}{2012^{2013}+1}\)
\(\Rightarrow2012.A=\dfrac{2012^{2013}+2012}{2012^{2013}+1}=\dfrac{2012^{2013}+1+2011}{2012^{2013}+1}=1+\dfrac{2011}{2012^{2013}+1}\)Ta có:\(B=\dfrac{2012^{2011}+1}{2012^{2012}+1}\)
\(\Rightarrow2012.B=\dfrac{2012^{2012}+2012}{2012^{2012}+1}=\dfrac{2012^{2012}+1+2011}{2012^{2012}+1}=1+\dfrac{2011}{2012^{2012}+1}\)Vì\(\dfrac{2011}{2012^{2013}+1}< \dfrac{2011}{2012^{2012}+1}\)
\(\Rightarrow1+\dfrac{2011}{2012^{2013}+1}< 1+\dfrac{2011}{2012^{2012}+1}\)
\(\Rightarrow\dfrac{2012^{2012}+1}{2012^{2013}+1}< \dfrac{2012^{2011}+1}{2012^{2012}+1}\)
Vậy A<B
Ta có \(\frac{2012.2013}{2012.2013+1}\)và \(\frac{2013}{2012}\)
Vì \(\frac{2012.2013}{2012.2013+1}< 1< \frac{2013}{2012}\)
nên \(\frac{2012.2013}{2012.2013+1}< \frac{2013}{2012}\)
\(\frac{2012}{2013}\)và \(\frac{2011}{2012}\)
phàn bù của \(\frac{2012}{2013}\)là \(\frac{1}{2013}\)
phàn bù của \(\frac{2011}{2012}\)là \(\frac{1}{2012}\)
Vì \(\frac{1}{2012}>\frac{1}{2013}\Rightarrow\frac{2012}{2013}>\frac{2011}{2012}\)
Ta có : \(\frac{2012\cdot2013}{2012\cdot2013+1}< 1\)
\(\frac{2013}{2012}>1\)
\(\Rightarrow\frac{2012\cdot2013}{2012\cdot2013+1}< \frac{2013}{2012}\)
Có : \(\frac{2012}{2013}=1-\frac{2012}{2013}=\frac{2013}{2013}-\frac{2012}{2013}=\frac{1}{2013}\)
\(\frac{2011}{2012}=1-\frac{2011}{2012}=\frac{2012}{2012}-\frac{2011}{2012}=\frac{1}{2012}\)
Vì \(2013< 2012\)nên \(\frac{1}{2013}< \frac{1}{2012}\)hay \(\frac{2012}{2013}< \frac{2011}{2012}\)
a)Ta có:
\(\frac{2011}{2012}>\frac{1006}{2012}=\frac{1}{2};\frac{2012}{2013}>\frac{2012}{4024}=\frac{1}{2}\)
\(\Rightarrow\)\(\frac{2011}{2012}+\frac{2012}{2013}>\frac{1}{2}+\frac{1}{2}=1\)hay \(\frac{2011}{2012}+\frac{2012}{2013}>1\)
Ta có: \(2011+2012< 2012+2013\Rightarrow\frac{2011+2012}{2012+2013}< 1\)
Suy ra: A>B
b) \(\frac{7}{16}=\frac{1}{8}+\frac{5}{16}=\frac{3}{16}+\frac{1}{4}=....\)