Bạn chép thiếu đầu bài rồi đáng nhẽ là : 

1x2+2x3+3x4+ .........+28x29+29x30

Ax3= 1x2x3 + 2x3x3+.....+29x30x3
lấy B= 1x2x3+ 2x3x4+......+29x30x31
B- Ax3= 0 + 1x2x3+ 2x3x4+ 28x29x30= B- 29x30x31. Suy ra Ax3= 29x30x31

Ax3= 1x2x3 + 2x3x3+.....+29x30x3 lấy B= 1x2x3+ 2x3x4+......+29x30x31 B- Ax3= 0 + 1x2x3+ 2x3x4+ 28x29x30= B- 29x30x31. Suy ra Ax3= 29x30x31

\(3S=1\times2\times3+2\times3\times3+...+29\times30\times3\\ 3S=1\times2\times\left(3-0\right)+2\times3\times\left(4-1\right)+...+29\times30\times\left(31-28\right)\\ 3S=1\times2\times3-1\times2\times3+2\times3\times4-...-28\times29\times30+29\times30\times31\\ 3S=29\times30\times31\\ S=29\times10\times31=8990\)

S = 333300 NHA , BẠN CÓ CẦN CÁCH GIẢI KO

S = 1.2+2.3+3.4+......+99.100 
Gấp S lên 3 lần ta có: 
S . 3 = 1.2.3 + 2.3.3 + 3.4.3 + … + 99.100.3 
S . 3 = 1.2.3 + 2.3.(4 - 1) + 3.4.( 5 - 2) + … + 99.100. (101 - 98) 
S . 3 = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + … + 99.100.101 - 98.99.100 
S . 3 = 99.100.101 
S = 99.100.101 : 3 
S = 33.100.101 
S = 333 300

Gọi biểu thức trên là A, ta có :

A = 1x2 + 2x3 + 3x4 + 4x5 + ...+ 99x100

A x 3 = 1x2x3 + 2x3x3 + 3x4x3 + 4x5x3 + ... + 99x100x3

A x 3 = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + 4x5x(6-3) + ... + 99x100x(101-98)

A x 3 = 1x2x3 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + 4x5x6 - 3x4x5 + ... + 99x100x101 - 98x99x100.

A x 3 = 99x100x101

A = 99x100x101 : 3

A = 333300

1x 2 + 2 x 3 + 3 x 4 + ...+ 99 x 100

Ta có: 

1 x 2 x 3                                           = 1 x 2 x 3

2 x 3 x 3     = 2 x 3 x ( 4 - 1)             = 2 x 3 x 4 -  1 x 2 x 3

3 x 4 x 3     =  3 x 4 x ( 5 - 2)            = 3 x 4 x 5 -   2 x 3 x 4 

........................................................= ........................................

99 x 100 x 3 = 99 x 100 x (101 - 98)  = 99 x 100 x 101 - 99 x 100 x 98

Cộng vế với vế ta có:

1 x 2 x 3 + 2 x 3 x 3 + 3 x 4 x 3 +...+ 99 x 100 x 3 = 99 x100 x 101

(1 x 2 + 2 x 3 + 3 x 4 +...+ 99 x 100) x 3 =  99 x 100 x 101 

1 x 2 + 2 x 3 + 3 x 4 +...+ 99 x 100 = \(\dfrac{99\times100\times101}{3}\)

1 x 2 + 2 x 3 + 3 x 4 + ....+ 99 x 100 = 333300

333300

\(S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2017}-\frac{1}{2018}\)

\(S=1-\frac{1}{2018}\)

\(S=\frac{2018}{2018}-\frac{1}{2018}\)

\(S=\frac{2017}{2018}\)

\(S=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2017.2018}.\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-...-\frac{1}{2017}+\frac{1}{2017}-\frac{1}{2018}\)

\(=1-\frac{1}{2018}=\frac{2017}{2018}\)