\(1.2.3....2015-1.2.3....2014-1.2.3....2013.2014^2\)

\(=1.2.3...\left(2014+1\right)-1.2.3...\left(2014+1\right)\)

\(=0\)

bạn làm thế là sai rồi 

có 3 con 2014 cơ mà 

Ta có : 

\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{2004.2005}\)

\(=\)\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2004}-\frac{1}{2005}\)

\(=\)\(\frac{1}{2}-\frac{1}{2005}\)

\(=\)\(\frac{2005}{4010}-\frac{2}{4010}\)

\(=\)\(\frac{2003}{4010}\)

Chúc bạn học tốt ~ 

Gọi \(A=\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{2014\cdot2015}\)

\(A=\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{2014\cdot2015}\)

\(A=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2014}-\frac{1}{2015}\)

\(A=\frac{1}{2}-\frac{1}{2015}\)

\(A=\frac{2015}{4030}-\frac{2}{4030}\)

\(A=\frac{2013}{4030}\)

chị kết bạn với em nha gửi lời kết bn với em nhé

j zậy em hả 

bằng 15 hay sao ý

Gọi phân số đã cho là \(A\).

Ta có:

\(A=\dfrac{1\times2015+2\times2014+3\times2013+\cdots+2015\times1}{1\times2+2\times3+3\times4+\cdots+2015\times2016}\)

Tử số:

Ta nhận thấy:

\(1\times2015+2\times2014+\cdots+2015\times1\)

\(=1(2016-1)+2(2016-2)+\cdots+2015(2016-2015)\)

\(=2016(1+2+\cdots+2015)-(1^2+2^2+\cdots+2015^2)\).

Mẫu số:

\(1\times2+2\times3+\cdots+2015\times2016\)

\(=1(1+1)+2(2+1)+\cdots+2015(2015+1)\)

\(=(1^2+2^2+\cdots+2015^2)+(1+2+\cdots+2015)\).

Đặt \(S=1+2+\cdots+2015\), \(T=1^2+2^2+\cdots+2015^2\).

Khi đó:

\(A=\dfrac{2016S-T}{T+S}\).

Với \(S=\dfrac{2015\times2016}{2}\),

\(T=\dfrac{2015\times2016\times4031}{6}\).

Thay vào:

\(A=\dfrac{2016\times\dfrac{2015\times2016}{2}-\dfrac{2015\times2016\times4031}{6}}{\dfrac{2015\times2016\times4031}{6}+\dfrac{2015\times2016}{2}}\)

Rút gọn \(2015\times2016\):

\(A=\dfrac{2016/2-4031/6}{4031/6+1/2}\)

\(=\dfrac{6048-4031}{4031+3}\)

\(=\dfrac{2017}{4034}=\dfrac12\).

=1/2 x 2/3 x 3/4 x 4/5 x 5/6 x.....x2013/2014 - 2014/2015

=1/2014 - 2014/2015

=1/2015

\(B=\left(\dfrac{1}{2015}+1\right)+\left(\dfrac{2}{2014}+1\right)+\left(\dfrac{3}{2013}+1\right)+...+\left(\dfrac{2014}{2}+1\right)+1\)

\(=\dfrac{2016}{2}+\dfrac{2016}{3}+...+\dfrac{2016}{2016}\)

=>B:A=2016