Bổ sung đề: So sánh A và B

Ta có:

A. \(2010^{1000}=\frac{1010^{1010}.2010^{1000}}{2010^{2010}}=\left(\frac{101}{201}\right)^{1010}\)

B. \(2010^{1000}=\frac{2010^{2010}.2010^{1000}}{3010^{3010}}=\left(\frac{201}{301}\right)^{3010}\)

Từ \(\frac{101}{201}>\frac{1}{2}>\frac{40401}{90601}=\left(\frac{201}{301}\right)^2\)và \(\frac{201}{301}< 1\)

  có: \(\left(\frac{101}{201}\right)^{1010}>\left(\frac{201}{301}\right)^{2.1010}=\left(\frac{201}{301}\right)^{2020}>\left(\frac{201}{301}\right)^{3010}\)

Suy ra \(A=\left(\frac{101}{201}\right)^{1010}.\frac{1}{2010^{1000}}>\left(\frac{201}{301}\right)^{3010}.\frac{1}{2010^{1000}}\) hay A > B

Ta có: \(3^{2010}=3^{10}\cdot3^{2000}=3^{10}\cdot9^{1000}\)

\(2^{3010}=2^{10}\cdot2^{3000}=2^{10}\cdot8^{1000}\)

Xét thấy \(3^{10}>2^{10};9^{1000}>8^{1000}\)

\(\Rightarrow3^{10}\cdot9^{1000}>2^{10}\cdot8^{1000}\)

Vậy \(3^{2010}>2^{3010}\)

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Vì 1000/2009>1000/2009+2010 (1)

     1010/2010>1010/2009+2010  (2)

  Ta cộng theo vế (1) và (2) với nhau nên ta được:

    1000/2009+1010/2010>1000/2009+2010 +1010/2009+2010

=>1000/2009+1010/2010>1000+1010/2009+2010

Vậy A<B

Chắc chắn 100% luôn, không sai đâu!!!!!!!

bằng nhau vì 2 biểu thức giống nhau

A=2.998508205

B=0.999502735

suy ra A>B

                                              Bài giải

Theo bài ra :  

\(A=\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2012}\)

\(B=\frac{2009+2010+2011}{2010+2011+2012}=\frac{2009}{2010+2011+2012}+\frac{2010}{2010+2011+2012}+\frac{2011}{2010+2011+2012}\)

Ta có : 

\(\frac{2009}{2010}>\frac{2009}{2010+2011+2012}\)

\(\frac{2010}{2011}>\frac{2010}{2010+2011+2012}\)

\(\frac{2011}{2012}>\frac{2011}{2010+2011+2012}\)

\(\Rightarrow\text{ }\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2012}>\frac{2009}{2010+2011+2012}+\frac{2010}{2010+2011+2012}+\frac{2011}{2010+2011+2012}\)

\(\Rightarrow\text{ }A>B\)

                                              Bài giải

Theo bài ra :  

\(A=\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2012}\)

\(B=\frac{2009+2010+2011}{2010+2011+2012}=\frac{2009}{2010+2011+2012}+\frac{2010}{2010+2011+2012}+\frac{2011}{2010+2011+2012}\)

Ta có : 

\(\frac{2009}{2010}>\frac{2009}{2010+2011+2012}\)

\(\frac{2010}{2011}>\frac{2010}{2010+2011+2012}\)

\(\frac{2011}{2012}>\frac{2011}{2010+2011+2012}\)

\(\Rightarrow\text{ }\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2012}>\frac{2009}{2010+2011+2012}+\frac{2010}{2010+2011+2012}+\frac{2011}{2010+2011+2012}\)

\(\Rightarrow\text{ }A>B\)

SBT toán 6

\(1-A=1-\frac{2010^{2011}+1}{2010^{2012}+1}=\frac{2010^{2012}+1}{2010^{2012}+1}-\frac{2010^{2011}+1}{2010^{2012}+1}\)=\(\frac{2010}{2010^{2012}+1}\)

\(1-A=1-\frac{2010^{2011}+1}{2010^{2012}+1}=\frac{2010^{2012}+1}{2010^{2012}+1}-\frac{2010^{2011}+1}{2010^{2012}+1}=\frac{2010}{2010^{2012}+1}\)

\(1-B=1-\frac{2010^{2010}+1}{2010^{2011}+1}=\frac{2010^{2011}+1}{2010^{2011}+1}-\frac{2010^{2010}+1}{2010^{2011}+1}=\frac{2010}{2010^{2011}+1}\)

\(\frac{2010}{2010^{2012}+1}<\frac{2010}{2010^{2011}+1}\Rightarrow A>B\)

Ta có:

\(2010A=\frac{2010^{2012}+2010}{2010^{2012}+1}=\frac{2010^{2012}+1+2009}{2010^{2012}+1}=1+\frac{2009}{2010^{2012}+1}\) 

\(2010B=\frac{2010^{2011}+2010}{2010^{2011}+1}=\frac{2010^{2011}+1+2009}{2010^{2011}+1}=1+\frac{2009}{2010^{2011}+1}\)

Do \(2010^{2012}+1>2010^{2011}+1\) => \(\frac{2009}{2010^{2012}+1}<\frac{2009}{2010^{2011}+1}\)

Nên \(1+\frac{2009}{2010^{2012}+1}<1+\frac{2009}{2010^{2011}+1}\) hay 2010A < 2010B

Vậy A<B

\(1-A=1-\frac{2010^{2011}+1}{2010^{2012}+1}=\frac{2010^{2012}+1}{2010^{2012}+1}-\frac{2010^{2011}+1}{2010^{2012}+1}=\frac{2010}{2010^{2012}+1}\)

\(1-B=1-\frac{2010^{2010}+1}{2010^{2011}+1}=\frac{2010^{2011}+1}{2010^{2011}+1}-\frac{2010^{2010}+1}{2010^{2011}+1}=\frac{2010}{2010^{2011}+1}\)

Do \(\frac{2010}{2010^{2012}+1}B\)

Do 2010²⁰¹¹+1<2010²⁰¹²+1=>A<1

Tương tự với B;B<1

Theo đề bài ta có:

\(A=\frac{2010^{2011}+1}{2010^{2012}+1}

A>B bạn nha

CHÚC BẠN HỌC GIỎI

\(B=\frac{2008+2009+2010}{2009+2010+2011}\)

\(=\frac{2008}{2009+2010+2011}+\frac{2009}{2009+2010+2011}+\frac{2010}{2009+2010+2011}\)

\(< \frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}=A\)

\(B=\frac{2008+2009+2010}{2009+2010+2011}\)

\(=\frac{2008}{2009+2010+2011}=\frac{2009}{2009+2010+2011}=\frac{2010}{2009+2010+2011}\)

\(< A=\frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}\)

A=\(\frac{-199}{10^{2011}}\)

B=\(\frac{-109}{10^{2011}}\)

Dễ dàng so sánh được A<B

A=-9/10²⁰¹¹+(-19/10²⁰¹⁰)

B=-9/10²⁰¹⁰+(-19/10²⁰¹¹)

Vì -9/10²⁰¹¹>(-19/10²⁰¹¹) và -9/10²⁰¹¹-(-19/10²⁰¹¹)=10/10²⁰¹¹

-19/10²⁰¹⁰<(-9/10²⁰¹⁰) và -9/10²⁰¹⁰-(-19/10²⁰¹⁰)=10/10²⁰¹⁰

mà 10/10²⁰¹¹<10/10²⁰¹⁰ nên suy ra B>A