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Ta có: \(B=\frac{2009^{2009}+1}{2009^{2010}+1}<\frac{2009^{2009}+1+2008}{2009^{2010}+1+2008}\)
\(=\frac{2009^{2009}+2009}{2009^{2010}+2009}\)
\(=\frac{2009.\left(2009^{2008}+1\right)}{2009.\left(2009^{2009}+1\right)}\)
\(=\frac{2009^{2008}+1}{2009^{2009}+1}=A\)
=> B<A
Ai k mik mik k lại. Chúc các bạn thi tốt
Ta có: $B=\frac{2009^{2009}+1}{2009^{2010}+1}<\frac{2009^{2009}+1+2008}{2009^{2010}+1+2008}$B=20092009+120092010+1 <20092009+1+200820092010+1+2008
$=\frac{2009^{2009}+2009}{2009^{2010}+2009}$=20092009+200920092010+2009
$=\frac{2009.\left(2009^{2008}+1\right)}{2009.\left(2009^{2009}+1\right)}$=2009.(20092008+1)2009.(20092009+1)
$=\frac{2009^{2008}+1}{2009^{2009}+1}=A$=20092008+120092009+1 =A
=> B<A
Ai k mik mik k lại. Chúc các bạn thi tốt
\(B=\frac{2008+2009+2010}{2009+2010+2011}\)
\(=\frac{2008}{2009+2010+2011}+\frac{2009}{2009+2010+2011}+\frac{2010}{2009+2010+2011}\)
\(< \frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}=A\)
Ta có :
\(2009A=\frac{2009\left(2009^{2008}+1\right)}{2009^{2009}+1}=\frac{2009^{2009}+2009}{2009^{2009}+1}=1+\frac{2008}{2009^{2009}+1}\)
\(2009B=\frac{2009\left(2009^{2009}+1\right)}{2009^{2010}+1}=\frac{2009^{2010}+2009}{2009^{2010}+1}=1+\frac{2008}{2009^{2010}+1}\)
Vì \(1+\frac{2008}{2009^{2009}+1}>1+\frac{2008}{2009^{2010}+1}\Rightarrow2009A>2009B\)
=> A > B
Vì B là phân số bé hơn 1 nên cộng cùng một số vào tử và mẫu của phân số đó thì giá trị của B sẽ tăng thêm, ta có:
\(B=\frac{2009^{2009}+1}{2009^{2010}+1}< \frac{2009^{2009}+1+2008}{2009^{2010}+1+2008}=\frac{2009^{2009}+2009}{2009^{2010}+2009}=\frac{2009\left(2009^{2008}+1\right)}{2009\left(2009^{2009}+1\right)}=\frac{2009^{2008}+1}{2009^{2009}+1}=A\)
Vậy B < A
\(B=\frac{2009^{2010}-2}{2009^{2011}-2}< 1\)
\(\Rightarrow B=\frac{2009^{2010}-2}{2009^{2011}-2}< \frac{2009^{2010}-2+2011}{2009^{2011}-2+2011}=\frac{2009^{2010}+2009}{2009^{2011}+2009}\)\(=\frac{2009.\left(2009^{2009}+1\right)}{2009.\left(2009^{2010}+1\right)}=\frac{2009^{2009}+1}{2009^{2010}+1}\)
Suy ra : \(\frac{2009^{2010}-2}{2009^{2011}-2}< \frac{2009^{2009}+1}{2009^{2010}+1}\) hay \(B< A\)
Vậy \(A>B\)
Ta có:
\(\frac{2009^{2008+1}}{2009^{2009+1}}=\frac{2009^{2009}}{2009^{2010}}=\frac{1}{2009}\)
\(\frac{2009^{2008+5}}{2009^{2009+9}}=\frac{2009^{2013}}{2009^{2018}}=\frac{1}{2009^5}\)
=>Đẳng thức trên lớn hơn đẳng thức dứi(vì 2009<2009^5)
Vậy.......
1.
\(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{99}}+\frac{1}{2^{100}}+\frac{1}{2^{100}}\)
\(=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}+\left(\frac{1}{2^{100}}+\frac{1}{2^{100}}\right)\)
\(=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}+\frac{1}{2^{99}}\)
cứ làm như vậy ta được :
\(=1+1=2\)
2. Ta có :
\(\frac{2008+2009}{2009+2010}=\frac{2008}{2009+2010}+\frac{2009}{2009+2010}\)
vì \(\frac{2008}{2009}>\frac{2008}{2009+2010}\); \(\frac{2009}{2010}>\frac{2009}{2009+2010}\)
\(\Rightarrow\frac{2008}{2009}+\frac{2009}{2010}>\frac{2008+2009}{2009+2010}\)
B = 20092009 + 1 / 20092010+1 < 20092009+1+2008 / 20092010+1+2008
= 20092009+2009 / 20092010+2009
= 2009(20092008+1) / 2009(20092009+1)
= 20092008+1 / 20092009+1 = A
=> A > B nhé!
Ai k mk mk k lại !!
Vậy bạn phả xét bổ đề \(\frac{a}{b}<\frac{a+n}{b+n}\)