Lời giải:

$(1.2+2.3+3.4+...+2012.2013)-(2^2+3^2+...+2013^2)$

$=[(2-1).2+(3-1).3+(4-1).4+...+(2013-1).2013]-(2^2+3^2+...+2013^2)$

$=(2^2+3^2+4^2+...+2013^2)-(2+3+4+...+2013)-(2^2+3^2+...+2013^2)$

$=-(2+3+4+...+2013)$

$=1-(1+2+3+...+2013)$

$=1-2013.2014:2=1-2027091=-2027090$

(1.2+2.3+3.4+.....+2012.2013)-(2²+3²+4²+......+2013²)

= 1.2 + 2.3 + 3.4 +...+ 2012.2013 - 2² -3² - 4² -....-2013²

=1.2 + 2.3 + 3.4 + ...+2012.2013 - 2.2 -3.3 - 4.4 -...- 2013.2013

=(1.2 - 2.2) + (2.3 - 3.3) + (3.4 - 4.4) + ...+(2012.2013 - 2013.2013)

=2.(1-2) + 3.(2-3) + 4.(3-4) +...+2013.(2012-2013)

=2.(-1) + 3.(-1) + 4.(-1) + ...+2013.(2012-2013)

= -2 - 3 - 4 -...- 2013

= -(2+3+4+...+2013)

= -[(2013+2).2012:2]

=-2027090

Ta có : 
A = 1.2 + 2.3 + 3.4 + ... + 198.199 + 199.200 
= 1.(1 + 1) + 2.(2 + 1) + 3.(3 + 1) + ... + 198(198 + 1) + 199(199 + 1) 
= (1^2 + 1) + (2^2 + 2) + (3^2 + 3) + ... + (198^2 + 198) + (199^2 + 199) 
= (1 + 2 + 3 + 4....+ 198 + 199) + (1^2 + 2^2 + 3^2 + ...+ 198^2 + 199^2) 
* Dễ chứng minh : 
....1 + 2 + 3 +...+ n = n(n + 1)/2 
.... 1^2 + 2^2 +...+ n^2 = [n(n + 1)(2n + 1)]/6 
Suy ra : A = [199.(199 + 1)]/2 + [199.(199 + 1)(2.199 + 1)]/6 = 2666600

ta có công thức 1.2+2.3+3.4+...+n.(n+1)=\(\frac{n\left(n+1\right)\left(n+2\right)}{3}\)

áp dụng công thức vào bài ta có: 1.2+2.3+3.4+...+2002.2003 = \(\frac{2002.2003.2004}{3}=2678684008\)

(1.2 + 2.3 + 3.4 + ... + 2012.2013) - (2² + 3² + 4² + 5² + ... + 2013²)

= [(2 - 1).2 + (3 - 1).3 + (4 - 1).4 + ... + (2013 - 1).2013] - (2² + 3² + 4² + 5² + ... + 2013²)

= (2² - 2 + 3² - 3 + 4² - 4 + ... + 2013² - 2013) - (2² + 3² + 4² + 5² + ... + 2013²)

= 2² - 2 + 3² - 3 + 4² - 4 + ... + 2013² - 2013 - 2² - 3² - 4² - 5² - ... - 2013²

= (2² - 2²) + (3² - 3²) + (4² - 4²) + ... + (2013² - 2013²) - (2 + 3 + 4 + ... + 2013)

= 0 - (2 + 3 + 4 + ... + 2013)

= 0 - (1 + 2 + 3 + 4 + ... + 2013) + 1

= 0 - \(\dfrac{2013.\left(2013+1\right)}{2}\) + 1

= 0 - 2027091 + 1

= (-2027091) + 1

= -2027090

 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

`S = 1.2 + 2.3 + 3.4 + 4.5 + ... + 99.100.`

`3S =  1.2.3 + 2.3.(4-1) + 3.4.(5-4) + 4.5.(6-3) + ... + 99.100.(101-98)`

`3S =  1.2.3 + 2.3.4-1.2.3 + 3.4.5-4.5.6 + 4.5.6-3.4.5 + ... + 99.100.101-98.99.100`

`3S =  99.100.101`

`S = 33.100.101`

`S = 333300`

3S=1.2(3-0)+2.3(4-1)+.....+99.100(101-98)

=1.2.3-0.1.2+2.3.4-1.2.3+4.5.6-2.3.4+....+99.100.101-98-99-100

=99.100.101

S=33.100.101

=333300