\(A=\left(1+\frac{1}{1.3}\right).\left(1+\frac{1}{2.4}\right).\left(1+\frac{1}{3.5}\right)...\left(1+\frac{1}{2011.2013}\right)\)

\(A=\frac{4}{1.3}.\frac{9}{2.4}.\frac{16}{3.5}...\frac{4048144}{2011.2013}\)

\(A=\frac{2.2}{1.3}.\frac{3.3}{2.4}.\frac{4.4}{3.5}...\frac{2012.2012}{2011.2013}\)

\(A=\frac{2.3.4...2012}{1.2.3...2011}.\frac{2.3.4...2012}{3.4.5...2013}\)

\(A=2012.\frac{2}{2013}=\frac{4024}{2013}\)

Ta có :

\(A=\left(1+\dfrac{1}{1.3}\right)\left(1+\dfrac{1}{2.4}\right)\left(1+\dfrac{1}{3.5}\right).................\left(1+\dfrac{1}{2011.2013}\right)\)

\(A=\left(\dfrac{3}{3}+\dfrac{1}{3}\right)\left(\dfrac{8}{8}+\dfrac{1}{8}\right)................\left(\dfrac{9999}{9999}+\dfrac{1}{9999}\right)\)

\(A=\dfrac{4}{3}.\dfrac{9}{8}...............\dfrac{10000}{9999}\)

\(A=\dfrac{4.9.................10000}{3.8.............9999}\)

\(A=\dfrac{2.2.3.3................100.100}{1.3.2.4...............99.101}\)

\(A=\dfrac{2.100}{101}=\dfrac{200}{101}\)

~ Chúc bn học tốt ~

\(\dfrac{1}{2001.2003}+\dfrac{1}{2003.2005}+...+\dfrac{1}{2011.2013}\)

\(=\dfrac{1}{2}\left(\dfrac{2}{2001.2003}+\dfrac{1}{2003.2005}+...+\dfrac{1}{2011.2013}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{1}{2001}-\dfrac{1}{2003}+\dfrac{1}{2003}-\dfrac{1}{2005}+...+\dfrac{1}{2011}-\dfrac{1}{2013}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{1}{2001}-\dfrac{1}{2013}\right)=\dfrac{2}{1342671}\)

=1/2.(2/2001.2003+2/2003.2005+.....+2/2011.2013)

=1/2(1/2001-1/2003+1/2003-1/2005+....+1/2011-1/2013)

=1/2(1/2001-1/2013)

=2/1342671

A= (1+1/1 x 3)x(1+1/2x4)x(1+1/3x5)x............x(1+1/2011x2013)

\(=\left(\frac{3}{3}+\frac{1}{3}\right)\left(\frac{8}{8}+\frac{1}{8}\right)....\left(\frac{4048143}{4048143}+\frac{1}{4048143}\right)\)

\(=\frac{4}{3}\cdot\frac{9}{8}\cdot...\cdot\frac{4048144}{4048143}\)

\(=\frac{4\cdot9\cdot....\cdot4048144}{3\cdot8\cdot....\cdot4048143}\)

\(=\frac{2\cdot2\cdot3\cdot3\cdot....\cdot2012\cdot2012}{1\cdot3\cdot2\cdot4\cdot....\cdot2011\cdot2013}\)

\(=\frac{2\cdot2012}{2013}=\frac{4024}{2013}\)

\(A=\frac{2012\times2013-1}{2011\times2013+2012}\)

\(A=\frac{2012\times2013-1}{2011\times2013+2011+1}\)

\(A=\frac{2012\times2013-1}{2013\times\left(2011+1\right)+1}\)

\(A=\frac{2012\times2013-1}{2013\times2012+1}\)

\(A=\frac{2012\times2013+1-2}{2012\times2013+1}\)

\(A=1-\frac{2}{2012\times2013+1}\)

Tính tử số 2012x2013-1=2011x2013+2013-1=2011x2013+2012

Do tử=mẫu =2011x2013=2012

Suy ra: A=1

\(A=\frac{2012\times2013-1}{2011\times2013+2012}\)

\(A=\frac{\left(2011+1\right)\times2013-1}{2011\times2013+2012}=\frac{2011\times2013+2013-1}{2011\times2013+2012}=\frac{2011\times2013+2012}{2011\times2013+2012}=1\)

Trả lời

2012.2013-1/2011.2013+2012

=2011.2013+2012-1/2011.2013+2012

=-1/1

=-1.

Học tốt !

\(\frac{2012.2013-1}{2011.2013+2012}\)

\(=\frac{2012.2013-1}{\left(2012-1\right)\left(2012+1\right)+2012}\)

\(=\frac{2012.2013-1}{2012^2-1^2+2012}\)

\(=\frac{2012.2013-1}{2012.2013-1}\)

\(=1\)

Chúc em học tốt !!!

ta có

        (2011+1)+(2013-1)

         =(2012-1)

vì (2012-1)<2012²

=>2011x2013<2012²

Ta có :

2011 . 2013

= 2011 . ( 2012 + 1 ) 

= 2011 . 2012 + 2011 . 1

2012² 

= 2012 . 2012

= ( 2011 + 1 ) . 2012

= 2011 . 2012 + 2012 . 1

Ta thấy : 2011 . 1 < 2012 . 1 nên 2011 . 2012 + 2011 . 1 < 2011 . 2012 + 2012 . 1

Vậy : 2011 . 2013 < 2012²

2011 x 2013 bé hơn