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a) Ta có : \(\frac{2010}{2011}>\frac{2010}{2011+2012}\)
\(\frac{2011}{2012}>\frac{2011}{2011+2012}\)
Nên \(\frac{2010}{2011}+\frac{2011}{2012}>\frac{2010+2011}{2011+2012}\)=> M > N
b) P = \(\frac{2011.2012-2}{2010.2011+4020}=\frac{2011.\left(2010+2\right)-2}{2010.2011+4020}=\frac{2011.2010+2011.2-2}{2010.2011+4020}=\)\(\frac{2011.2010+4020}{2010.2011+4020}=1\)
Nên P = 1
câu b sửa lại:\(P=\frac{2011.2012-2}{2010.2011+4020}=\frac{2011.2010+4022-2}{2010.2011+4020}=\frac{2010.2011+4020}{2010.2011+4020}=1\)
chắc chắn là A > B
hãy ủng hộ mk bằng một niềm tin nhé
^ _ ^ hihi
\(A=\frac{2011\times2012}{2011+2012}+\frac{2009\times2010}{2009+2010}\)
\(A=\frac{2011\times2011}{2011+2012}+\frac{2011}{2011+2012}+\frac{2010\times2010}{2009+2010}-\frac{2010}{2009+2010}\)
\(A=\frac{2011\times2011}{2011+2012}+\frac{2010\times2010}{2009+2010}+\frac{2011}{2011+2012}-\frac{2010}{2009+2010}\)
\(A=B+\frac{2011}{2011+2012}-\frac{2010}{2009+2010}\)
\(A=B+\frac{2011}{4023}-\frac{2010}{4019}\)
Dễ thấy \(\frac{2011}{4023}-\frac{2010}{4019}< 0\)
\(\Rightarrow A< B\)
#)Giải :
Ta có : \(1-\frac{2010}{2011}=\frac{1}{2011}\)
\(1-\frac{2011}{2012}=\frac{1}{2012}\)
Vì \(\frac{1}{2011}>\frac{1}{2012}\Rightarrow\frac{2010}{2011}>\frac{2011}{2012}\)
\(\frac{2010}{2011}=1-\frac{1}{2011}\)
\(\frac{2011}{2012}=1-\frac{1}{2012}\)
\(2011\)<\(2012\)\(\Rightarrow\frac{1}{2011}\)>\(\frac{1}{2012}\)
\(\Rightarrow\frac{2010}{2011}\)<\(\frac{2011}{2012}\)
\(\dfrac{2010}{2011}+\dfrac{2011}{2012}+\dfrac{2012}{2013}+\dfrac{2013}{2011}\)
=1-\(\dfrac{1}{2011}\)+1\(-\dfrac{1}{2012}\)+1-\(\dfrac{1}{2013}\)+1-\(\dfrac{1}{2011}\)
=4-(\(\dfrac{2}{2011}+\dfrac{1}{2012}+\dfrac{1}{2013}\)) < 4
m=\(\dfrac{2010}{2011}+\dfrac{2011}{2012}+\dfrac{2012}{2013}+\dfrac{2013}{2011}\)
=\(1-\dfrac{1}{2011}+1-\dfrac{1}{2012}+1-\dfrac{1}{2013}+1+\dfrac{2}{2011}\)
=4+\(\dfrac{1}{2011}-\dfrac{1}{2012}-\dfrac{1}{2013}\)
vì:
do \(\dfrac{1}{2011}< 1\)
\(\dfrac{1}{2012}< 1\)
\(\dfrac{1}{2013}< 1\)
nên \(\dfrac{1}{2011}-\dfrac{1}{2012}-\dfrac{1}{2013}< 1-1-1=-1\)
hay \(\dfrac{1}{2011}-\dfrac{1}{2012}-\dfrac{1}{2013}< 0\)
nên 4+\(\dfrac{1}{2011}-\dfrac{1}{2012}-\dfrac{1}{2013}< 4\)
vậy tổng m <4
bài này mình tưởng phải lên cấp 2 mới có thế mà mấy em lớp 4 đã phải làm á
\(M=\frac{2010}{2011}+\frac{2011}{2012}>\frac{2010}{2011+2012}+\frac{2011}{2011+2012}=\frac{2010+2011}{2011+2012}=N\)