Tìm max của biểu thức: 1 3 4 2 + − x x .

con cho

To quá

\(A=\frac{2006}{2007}+\frac{2007}{2008}+\frac{2008}{2009}=1-\frac{1}{2007}+1-\frac{1}{2008}+1-\frac{1}{2009}\)

\(=3-\frac{1}{2007}-\frac{1}{2008}-\frac{1}{2009}>1\).

\(B=\frac{2006+2007+2008}{2007+2008+2009}< \frac{2007+2008+2009}{2007+2008+2009}=1\).

Suy ra \(A>B\).

P/s : Lớp 6 nhé bạn 

Dấu \(.\)là dấu nhân 

Đặt \(A=\frac{2008}{1}+\frac{2007}{2}+\frac{2006}{3}+...+\frac{2}{2007}+\frac{1}{2008}\)  

      \(B=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2008}+\frac{1}{2009}\)

Ta có : 

\(A=\frac{2008}{1}+\frac{2007}{2}+\frac{2006}{3}+...+\frac{2}{2007}+\frac{1}{2008}\)

\(\Rightarrow A=1+\left(\frac{2007}{2}+1\right)+\left(\frac{2006}{3}+1\right)+...+\left(\frac{2}{2007}+1\right)+\left(\frac{1}{2008}+1\right)\)

\(\Rightarrow A=\frac{2009}{2009}+\frac{2009}{2}+\frac{2009}{3}+...+\frac{2009}{2007}+\frac{2009}{2008}\)

\(\Rightarrow A=\frac{2009}{2}+\frac{2009}{3}+...+\frac{2009}{2007}+\frac{2009}{2008}+\frac{2009}{2009}\)

\(\Rightarrow A=2009.\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2007}+\frac{1}{2008}+\frac{1}{2009}\right)\)

\(\Rightarrow A=2009.B\)

Nên : \(\frac{A}{B}=\frac{2009.B}{B}=2009\)

Vậy kết quả biểu thức đã cho là \(2009\)

~ Ủng hộ nhé 

\(\frac{\frac{2008}{1}+\frac{2007}{2}+\frac{2006}{3}+...+\frac{2}{2007}+\frac{1}{2008}}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2008}+\frac{1}{2009}}\)

\(=\frac{\left(\frac{2007}{2}+1\right)+\left(\frac{2006}{3}+1\right)+...+\left(\frac{2}{2007}+1\right)+\left(\frac{1}{2008}+1\right)+1}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2008}+\frac{1}{2009}}\)

\(=\frac{\frac{2009}{2}+\frac{2009}{3}+...+\frac{2009}{2007}+\frac{2009}{2008}+\frac{2009}{2009}}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2008}+\frac{1}{2009}}\)

\(=\frac{2009.\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2007}+\frac{1}{2008}+\frac{1}{2009}\right)}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2008}+\frac{1}{2009}}\)

\(=2009\)

\(\frac{\frac{2008}{1}+\frac{2007}{2}+...+\frac{1}{2008}}{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2008}+\frac{1}{2009}}\)

\(=\frac{\frac{1}{1}+\left(1+\frac{2007}{2}\right)+...+\left(1+\frac{1}{2008}\right)}{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2008}+\frac{1}{2009}}\)

\(=\frac{\frac{2009}{2}+\frac{2009}{3}+...+\frac{2009}{2009}}{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2009}}\)

\(=\frac{2009\times\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2009}\right)}{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2009}}\)

\(=2009\)

\(=\left(\dfrac{2007}{2}+1\right)+\left(\dfrac{2006}{3}+1\right)+...+\left(\dfrac{2}{2007}+1\right)+\left(\dfrac{1}{2008}+1\right)+1\)

\(=\dfrac{2009}{2}+\dfrac{2009}{3}+...+\dfrac{2009}{2008}+\dfrac{2009}{2009}\)

\(=2009\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2009}\right)\)