A = 1.2 + 2.3 + 3.4 + ... + 2013.2014 
3A = 1.2.3 + 2.3.3 + 3.4.3 + ... + 2013.2014.3 
Mà : 
1.2.3 = 1.2.3 
2.3.3 = 2.3.4 - 2.3.1 
3.4.3 = 3.4.5 - 3.4.2 

2012.2013.3 = 2012.2013.2014 - 2012.2013.2011 
2013.2014.3 = 2013.2014.2015 - 2013.2014.2012 
Cộng tất cả, vế theo vế ---> 3S = 2013.2014.2015 
---> A = 2013.2014.2015 / 3 = 2723058910.

của bạn đây

Lời giải:

$3A=1.2.3+2.3(4-1)+3.4(5-2)+...+2013.2014(2015-2012)$

$=(1.2.3+2.3.4+3.4.5+...+2013.2014.2015)-(1.2.3+2.3.4+...+2012.2013.2014)$

$=2013.2014.2015$

$\Rightarrow A=\frac{2013.2014.2015}{3}$

\(\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{2013.2014}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{2013}-\dfrac{1}{2014}\\ =1-\dfrac{1}{2014}\\ =\dfrac{2013}{2014}\)

a= 1.2 + 2.3 + 3.4 + 4.5 + ... + 2013.2014
3a = 1.2.3 +2.3.3 +3.4.3 +4.5.3 +... + 2013.2014.3
3a = 1.2.(3-0)+2.3(4-1)+3.4.(5-2)+...+2013.2014.(2015-2012)
3a = 1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+2013.2014.2015-2012.2013.2014
3a=2013.2014.2015
a = 2013.2014.20153
a = 2723058910

hình như bạn chép sai đề rùi,xem lại nhé!

S=1/1.2+1/2.3+1/3.4+...+1/2013.2014

S=1/1-1/2+1/2-1/3+1/3-1/4+...+1/2013-1/2014

S=1-1/2014

S=2013/2014

S=1-1/2+1/2-1/3+.....+1/2013-1/2014

S=1-1/2014

S=2013/2014  

hình như bạn chép đề sai.

S=1/1.2^₊1/2.3+1/3.4+...+1/2013.2014

S=1/1-1/2+1/2-1/3+1/3-1/4+...+1/2013-1/2014

S=1-1/2014

S=2013/2014

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

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

2013/2014

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

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

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

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

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