vì 1/1*2=1-1/2

   1/2*3=1/2-1/3

.....................

1/2014*2015=1/2014-1/2015

=1-1/2+1/2-1/3+1/3-....+1/2014-1/2015

=1-1/2015

=2014/2115

\(\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+....+\frac{1}{2014x2015}\)

=\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{99}-\frac{1}{100}\)

=\(1-\frac{1}{100}\)

=\(\frac{99}{100}\)

A=1*2+2*3+...+2014*2015

3A=1*2*3+2*3*(4-1)+...+2014*2015*(2016-2013)

3A=1*2*3+2*3*4-1*2*3+...+2014*2015*2016-2013*2014*2015

3A=2014*2015*2016

A=2014*2015*2016/3

A=2727117120

Gọi biểu thức trên là A, ta có :

A = 1x2 + 2x3 + 3x4 + 4x5 + ...+ 2014x2015

A x 3 = 1x2x3 + 2x3x3 + 3x4x3 + 4x5x3 + ... + 2014x2015x3

A x 3 = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + 4x5x(6-3) + ... + 2014x2015x(2016-2013)

A x 3 = 1x2x3 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + 4x5x6 - 3x4x5 + ... + 2014x2015x2016 - 2014x2015x2013.

A x 3 = 2014x2015x2016

A = 2014x2015x2016 : 3

A = 2727117120

\(\dfrac{1}{1\times2}+\dfrac{1}{2\times3}+\dfrac{1}{3\times4}+....+\dfrac{1}{24\times25}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{24}-\dfrac{1}{25}\)

\(=1-\dfrac{1}{25}\)

\(=\dfrac{24}{25}\)

Nhanh giúp mình với cả nhà ơi

Ta có : 

\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2013.2014}\)

\(=\)\(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2013}-\frac{1}{2014}\)

\(=\)\(1-\frac{1}{2014}\)

\(=\)\(\frac{2014}{2014}-\frac{1}{2014}\)

\(=\)\(\frac{2013}{2014}\)

Vậy \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2013.2014}=\frac{2013}{2014}\)

Dấu \(.\) là dấu nhân nhé 

Chúc bạn học tốt ~

\(\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{2013\times2014}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2013}-\frac{1}{2014}\)

\(=1-\frac{1}{2014}\)

\(=\frac{2013}{2014}\)

CHÚC BN HỌC TỐT!!!!!
 

\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+............+\frac{1}{2013.2014}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.........+\frac{1}{2013}-\frac{1}{2014}\)

\(=1-\frac{1}{2014}\)

\(=\frac{2013}{2014}\)

ai làm nhanh mik cho

mik cho

haizzz đáng tiếc tôi muốn ns là:  ko bao f và đừng mong chờ OK