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

bạn giải được không

3A=3.(1x2+2x3+3x4+....+99x100)

3A=3x1x2x+3x2x3+3x3x4+...+3x99x100

3A=(3-0)x1x2+(4-1)x2x3+(5-2)x3x4+...+(101-98)x99x100

3A=1x2x3-0+2x3x4-1x2x3+3x4x5-2x3x4+...+99x100x101-98x99x100

3A=99x100x101-0

3A=99x100x101

A=33x100x101

A=333 300

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 = 99x100x101 A = 333300

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 = 99x100x101
A = 333300

Đặt A=1.2+2.3+3.4+...+99.100

3A=1.2.3+2.3.(4-1)+3.4.(5-2)+...+99.100.(101-98)

3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100

3A=99.100.101

A=333300

A = 333 300 nhé bn

333300

**** mik nha bak

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

A=1x2+2x3+3x4+4x5+......+99x100+100x101

3A=1x2x(3-0)+2x3x(4-1)+3x4x(5-2)+4x5x(6-3)+...+99x100x(101-98)+100x101x(102-99)

3A=1x2x3-0x1x2+2x3x4-1x2x3+3x4x5-2x3x4+4x5x6-3x4x5+...+99x100x101-98x99x100+100x101x102-99x100x101

3A=(1x2x3+2x3x4+3x4x5+4x5x6+...+99x100x101+100x101x102)-(0x1x2+1x2x3+2x3x4+3x4x5+...+98x99x100+99x100x101)

3A=100x101x102

A=100x101x102:3

A=343400

A = 1x2 + 2x3 + 3x4 + 4x5 + ... + 99x100 + 100x101

3A = 1x2x(3-0) + 2x3x(4-1) + 3x4x(5-2) + 4x5x(6-3) + ... + 99x100x(101-98) + 100x101x(102-99)

3A = 1x2x3 - 0x1x2 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + 4x5x6 - 3x4x5 + ... + 99x100x101 - 98x99x100 + 100x101x102 - 99x100x101

3A = 100x101x102 - 0x1x2

3A = 100x101x102

A = 100x101x34

A = 343400

S = 1/100