Tính (2011*2010-1) / (2009*2011*2011+2010)
1 / 1 + 2009 / 2010 + 2009 / 2011 + 1 / 1 + 2000 / 2009 + 2010 / 2011 + 1 / 1 + 2011 / 2009 + 2011 /2010
tinh nhanh B=1/1+2009/2011+2009/2010 + 1/1+2010/2009+2010/2011 + 1/1+2011/2009+2011/2010
BT1: Tính
5) \(\dfrac{1}{1+\dfrac{2009}{2011}+\dfrac{2009}{2010}}+\dfrac{1}{1+\dfrac{2010}{2009}+\dfrac{2010}{2011}}+\dfrac{1}{1+\dfrac{2011}{2009}+\dfrac{2011}{2010}}\)
=\(\dfrac{1}{2009.\left(\dfrac{1}{2009}+\dfrac{1}{2011}+\dfrac{1}{2010}\right)}+\dfrac{1}{2010.\left(\dfrac{1}{2010}+\dfrac{1}{2009}+\dfrac{1}{2011}\right)}+\dfrac{1}{2011.\left(\dfrac{1}{2011}+\dfrac{1}{2009}+\dfrac{1}{2010}\right)}\)\(=\dfrac{1}{2009}:\left(\dfrac{1}{2009}+\dfrac{1}{2010}+\dfrac{1}{2011}\right)+\dfrac{1}{2010}:\left(\dfrac{1}{2009}+\dfrac{1}{2010}+\dfrac{1}{2011}\right)+\dfrac{1}{2011}:\left(\dfrac{1}{2009}+\dfrac{1}{2010}+\dfrac{1}{2011}\right)\)
\(=\left(\dfrac{1}{2009}+\dfrac{1}{2010}+\dfrac{1}{2011}\right):\left(\dfrac{1}{2009}+\dfrac{1}{2010}+\dfrac{1}{2011}\right)=1\)
So sánh :
A = 2009/2010 + 2010/2011 + 2011/2012
B = 2009 + 2010 + 2011/2010 + 2011 + 2012
Có : \(2009+2010>\dfrac{2009}{2010}\) ; \(2011+2012>\dfrac{2011}{2012}\)
\(\dfrac{2011}{2010}>1\) ; \(\dfrac{2010}{2011}< 1\) \(\Rightarrow\dfrac{2011}{2010}>\dfrac{2010}{2011}\)
Ta có : \(2009+2010+\dfrac{2011}{2010}+2011+2012>\dfrac{2009}{2010}+\dfrac{2010}{2011}+\dfrac{2011}{2012}\)
\(\Leftrightarrow B>A\)
Hay \(A< B\)
\(B=\frac{\frac{2008}{2011}+\frac{2009}{2010}+\frac{2010}{2009}+\frac{2011}{2008}+\frac{2012}{503}}{\frac{1}{2008}+\frac{1}{2009}+\frac{1}{2010}+\frac{1}{2011}}\)
2011 x 2010 - 1/2009 x 2011 + 2010
`(2011xx2020-1)/(2009xx2011+2010)`
`=((2009+1)xx2011-1)/(2009xx2011+2010)`
`=(2009xx2011+2011-1)/(2009xx2011+2010)`
`=(2009xx2011+2010)/(2009xx2011+2010)`
`=1`
\(\dfrac{2011.2010-1}{2009.2011+2010}\)
= \(\dfrac{2011.2009+2011-1}{2009.2011+2010}\)
= \(\dfrac{2011.2009+2010}{2009.2011+2010}\)
= 1
BT1: Tính:
\(\frac{1}{1+\frac{2009}{2009}+\frac{2009}{2010}}\) + \(\frac{1}{1+\frac{2010}{2009}+\frac{2010}{2011}}\) + \(\frac{1}{1+\frac{2011}{2009}+\frac{2011}{2010}}\)
so sanh
A=2008/2009+2009/2010+2010+2011
B=2008+2009+2010/2009+2010+2011
Dễ thấy:
\(\frac{2008}{2009}>\frac{2008}{2009+2010+2011}\)
\(\frac{2009}{2010}>\frac{2009}{2009+2010+2011}\)
\(\frac{2010}{2011}>\frac{2010}{2009+2010+2011}\)
=>\(\frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}>\frac{2008}{2009+2010+2011}+\frac{2009}{2009+2010+2011}+\frac{2010}{2009+2010+2011}\)
Hay \(\frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}>\frac{2008+2009+2010}{2009+2010+2011}\)
Vậy A > B
(2009 x 2010 + 2011 x 12 + 1998)/(2011 x 2010 - 2010 x 2009)