\(\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...+\frac{1}{\left(2n\right)^2}< \frac{1}{4}\)

\(=\frac{1}{2^2}\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}\right)< \frac{1}{4}\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{\left(n-1\right)n}\right)\)

\(=\frac{1}{4}\left(1-\frac{1}{n}\right)\)(đpcm)

Ta có:\(\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...+\frac{1}{\left(2n\right)^2}\)

\(=\frac{1}{4.4}+\frac{1}{4.9}+\frac{1}{4.16}+...+\frac{1}{4.n^2}\)

\(=\frac{1}{4}\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{n^2}\right)\)

\(Xét:\)

\(\frac{1}{2.2}< \frac{1}{1.2};\frac{1}{3.3}< \frac{1}{2.3};\frac{1}{4.4}< \frac{1}{3.4};\frac{1}{n.n}< \frac{1}{\left(n-1\right).n}...\)

\(Suyra:\)

\(P=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{n.n}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{\left(n-1\right).n}\)

\(\Leftrightarrow P< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{n-1}-\frac{1}{n}\)

\(\Leftrightarrow P< 1-\frac{1}{n}< 1\)

\(\Leftrightarrow\frac{1}{4}.P< 1.\frac{1}{4}\)

\(\Leftrightarrow\frac{1}{4}\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{n^2}\right)< \frac{1}{4}\)

\(\Leftrightarrow\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...+\frac{1}{\left(2n\right)^2}< \frac{1}{4}\left(đpcm\right)\)

Ta có : 

\(N=\frac{1}{4^2}+\frac{1}{6^2}+...+\frac{1}{\left(2n\right)^2}\)

\(N=\frac{1}{2^2}.\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}\right)\)

Ta thấy : \(\frac{1}{2^2}< \frac{1}{1.2}\)

\(\frac{1}{3^2}< \frac{1}{2.3}\)

.......

\(\frac{1}{n^2}< \frac{1}{\left(n-1\right).n}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< \frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{\left(n-1\right).n}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{n-1}-\frac{1}{n}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< 1-\frac{1}{n}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< 1\)

\(\Rightarrow\frac{1}{2^2}.\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}\right)< 1.\frac{1}{2^2}\)

\(\Rightarrow N< \frac{1}{4}\)(ĐPCM)

Ủng hộ mk nha !!! ^_^

bài này tui bít làm nhưng dài lắm

Cố gắng làm hộ mình với !

=>\(\frac{1}{2^2}\)x (\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{n^2}\))

Đặt A=\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{n^2}\)

Ta có:\(\frac{1}{2^2}\)<\(\frac{1}{1\cdot2}\)

         \(\frac{1}{3^2}\)<\(\frac{1}{2\cdot3}\)

.........\(\frac{1}{n^2}\)<\(\frac{1}{\left(n-1\right)\cdot n}\)

=>\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+...+\(\frac{1}{n^2}\)<\(\frac{1}{1\cdot2}\)+\(\frac{1}{2\cdot3}\)+...+\(\frac{1}{\left(n-1\right)\cdot n}\)

=>A<1-\(\frac{1}{2}\)+\(\frac{1}{2}\)-\(\frac{1}{3}\)+...+\(\frac{1}{n-1}\)--\(\frac{1}{n}\)
=>\(\frac{1}{2^2}\)*A<\(\frac{1}{2^2}\)(1--\(\frac{1}{n}\))

=>\(\frac{1}{2^2}\)*A<\(\frac{1}{4}\)(1--\(\frac{1}{n}\))

=>\(\frac{1}{2^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{\left(2n\right)^2}\)<\(\frac{1}{4}\)--\(\frac{1}{4n}\)<\(\frac{1}{4}\)

=>\(\frac{1}{2^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{\left(2n\right)^2}\)<\(\frac{1}{4}\)

Câu 1 :
 A = (2012+2) . [ ( 2012-2) : 3+1 ] : 2 = 2014 . 671 : 2 = 675697
 B = \(\frac{1}{2}\).  \(\frac{2}{3}\).  \(\frac{3}{4}\)+...+  \(\frac{2010}{2011}\).  \(\frac{2011}{2012}\)= \(\frac{1.2.3.....2010.2011}{2.3.4.....2011.2012}\)=  \(\frac{1}{2012}\)
Câu 2 :
 a) \(2x.\left(3y-2\right)+\left(3y-2\right)=-55\)
=> \(\left(3y-2\right).\left(2x+1\right)=-55\)
=>  \(3y-2;2x+1\in\: UC\left(-55\right)\)
=>  \(3y-2;2x+1=\left\{1;-1;5;-5;11;-11;55;-55\right\}\)
- Vậy ta có bảng 

\(\Leftrightarrow\)Những cặp (x;y) tìm được là : 
(-1;19)  ;   (2;-3)   ;    (5;-1)    ;    (-28;1)
b) Ta đặt vế đó là A
Ta xét A :   \(\frac{1}{4^2}\)<  \(\frac{1}{2.4}\)
                  \(\frac{1}{6^2}\)<  \(\frac{1}{4.6}\)
                  \(\frac{1}{8^2}\)<  \(\frac{1}{6.8}\)
                          ...
                 \(\frac{1}{\left(2n\right)^2}\)<  \(\frac{1}{\left(2n-2\right).2n}\)

  \(\Leftrightarrow\)A < \(\frac{1}{2.4}\)+  \(\frac{1}{4.6}\)+...+  \(\frac{1}{\left(2n-2\right).2n}\)
  \(\Leftrightarrow\)A < \(\frac{1}{2}\). ( \(\frac{2}{2.4}\)+  \(\frac{2}{4.6}\)+...+  \(\frac{2}{\left(2n-2\right).2n}\))
  \(\Leftrightarrow\)A < \(\frac{1}{2}\). ( \(\frac{1}{2}\)-  \(\frac{1}{4}\)+  \(\frac{1}{4}\)-  \(\frac{1}{6}\)+...+  \(\frac{1}{2n-2}\)-  \(\frac{1}{2n}\))
  \(\Leftrightarrow\)A < \(\frac{1}{2}\). ( \(\frac{1}{2}\)-  \(\frac{1}{2n}\)) = \(\frac{1}{2}\).  \(\frac{1}{2}\)-  \(\frac{1}{2}\).  \(\frac{1}{2n}\)
  \(\Leftrightarrow\)A < \(\frac{1}{4}\)-  \(\frac{1}{4n}\)<  \(\frac{1}{4}\) ( Vì n \(\in\)N )
  \(\Leftrightarrow\)A <  \(\frac{1}{4}\)( đpcm ) .

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