S =  1x2 + 2x3 + 3x4 + ……………… + 11x12 + 12x13
3S=1x2x3 + 2x3x3 + 3x4x3+ ………. + 11x12x3 + 12x13x3

Ta lấy K = 1x2x3 +2x3x4  + 3x4x5 + ……   + 11x12x13 + 12x13x14
 -     3S  = 1x2x3 + 2x3x3 + 3x4x3+ ………  + 11x12x3  + 12x13x3
            ------------------------------------------------------------------------------------
 K – 3S =     0     + 2x3x1  + 3x4x2 + …… .. + 11x12x10 + 12x13x11

K – 3S =   K – 12x13x14
Từ đó suy ra:   3S = 12x13x14
S = 4x13x14 = 728  
 

Cách 2:
S x 3 = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + …. + 11x12x(13-10) + 12x13x(14-11)
S x 3 = 1x2x3 + 2x3x4 – 2x3x1 + 3x4x5 – 3x4x2 + …..+ 11x12x13 – 11x12x10 +12x13x14 – 12x13x11
S x 3 = 12 x 13 x14
S = 4 x 13 x 14
S = 728 

3C = 1x2x3 + 2x3x(4-1) + ... + 11x12x(13-9)

3C = 1x2x3 + 2x3x4 - 1x2x3 + ... + 11x12x13 - 9x11x12

3C = 9x11x12

3C = 1188

C = 396

C = 1x2 + 2x3 + 3x4 + ... + 10x11 + 11x12

=> 3C = 1x2x3 + 2x3x4 - 1x2x3 + ... + 11x12x13 - 9x11x12

=> 3C = 9 x 11 x 12

=> 3C = 1188

=> C = 396

S =  1x2 + 2x3 + 3x4 + ……………… + 11x12 + 12x13
3S=1x2x3 + 2x3x3 + 3x4x3+ ………. + 11x12x3 + 12x13x3

Ta lấy K = 1x2x3 +2x3x4  + 3x4x5 + ……   + 11x12x13 + 12x13x14
 -     3S  = 1x2x3 + 2x3x3 + 3x4x3+ ………  + 11x12x3  + 12x13x3
            ------------------------------------------------------------------------------------
 K – 3S =     0     + 2x3x1  + 3x4x2 + …… .. + 11x12x10 + 12x13x11

K – 3S =   K – 12x13x14
Từ đó suy ra:   3S = 12x13x14
S = 4x13x14 = 728  

S=1x2 + 2x3 + 3x4 + ... + 11x12 + 12x13

3S=1x2x3 + 2x3x3 + 3x4x3 + ... + 11x12x13 

    =1x2x3 + 2x3x(4-1) + 3x4x(5-2) + ... + 11x12x(13-10)

    =1x2x3 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + ... + 11x12x13 - 10x11x12

    =\(\frac{10x11x12}{3}\) = 1320 

Lộn đề rùi Đây mới là bài giải đúng nè S=1x2 + 2x3 + 3x4 + ... + 11x12 + 12x133S= 1x2x3 + 2x3x3 + 3x4x3 + ... + 11x12x13 + 12x13x13    = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + ... + 11x12x(13-10) + 12x13x(14-11)    = 1x2x3 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + ... + 11x12x13 - 10x11x12 + 12x13x14 - 11x12x13    = (12x13x14) : 3          = 728                                      

S=728

Cách 1:
S =  1x2 + 2x3 + 3x4 + ……………… + 11x12 + 12x13
3S=1x2x3 + 2x3x3 + 3x4x3+ ………. + 11x12x3 + 12x13x3

Ta lấy K = 1x2x3 +2x3x4  + 3x4x5 + ……   + 11x12x13 + 12x13x14
 -     3S  = 1x2x3 + 2x3x3 + 3x4x3+ ………  + 11x12x3  + 12x13x3
            ------------------------------------------------------------------------------------
 K – 3S =     0     + 2x3x1  + 3x4x2 + …… .. + 11x12x10 + 12x13x11

K – 3S =   K – 12x13x14
Từ đó suy ra:   3S = 12x13x14
S = 4x13x14 = 728  

Cách 2:
S x 3 = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + …. + 11x12x(13-10) + 12x13x(14-11)
S x 3 = 1x2x3 + 2x3x4 – 2x3x1 + 3x4x5 – 3x4x2 + …..+ 11x12x13 – 11x12x10 +12x13x14 – 12x13x11
S x 3 = 12 x 13 x14
S = 4 x 13 x 14
S = 728 

Giải
Cách 1:
S =  1x2 + 2x3 + 3x4 + ……………… + 11x12 + 12x13
3S=1x2x3 + 2x3x3 + 3x4x3+ ………. + 11x12x3 + 12x13x3

Ta lấy K = 1x2x3 +2x3x4  + 3x4x5 + ……   + 11x12x13 + 12x13x14
 -     3S  = 1x2x3 + 2x3x3 + 3x4x3+ ………  + 11x12x3  + 12x13x3
            ------------------------------------------------------------------------------------
 K – 3S =     0     + 2x3x1  + 3x4x2 + …… .. + 11x12x10 + 12x13x11

K – 3S =   K – 12x13x14
Từ đó suy ra:   3S = 12x13x14
S = 4x13x14 = 728  
 

Cách 2:
S x 3 = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + …. + 11x12x(13-10) + 12x13x(14-11)
S x 3 = 1x2x3 + 2x3x4 – 2x3x1 + 3x4x5 – 3x4x2 + …..+ 11x12x13 – 11x12x10 +12x13x14 – 12x13x11
S x 3 = 12 x 13 x14
S = 4 x 13 x 14
S = 728 

Xét 3S= 1 x 2 x ( 3 - 0) + 2 x 3 x (4 -1) +....+ 11 x 12 x(13 -10 ) + 12 x13 x (14-11)

          = 1 x2 x 3 -1x2x3 +2 x3 x4 +.... -10x11x12 +11x 12 x13 -11x12 x13 +12x13x14

          =12 x 13 x14 ---> S=\(\frac{12\times13\times14}{3}=728\)

S = 728

Nguồn: Tính tổng: 1x2 + 2x3 + 3x4 +...+ 2019x2020 + 2020x2021 - Hoc24
Đặt $A=1.2+2.3+3.4+.........+2019.2020+2020.2021$

$\Rightarrow 3A=\mathrm{1.2.3}+\mathrm{2.3.3}+\mathrm{3.4.3}+.....+\mathrm{2019.2020.3}+\mathrm{2020.2021.3}$

$=\mathrm{1.2.3}+\mathrm{2.3.}\left(4-1\right)+\mathrm{3.4.}\left(5-2\right)+.....+\mathrm{2020.2021.}\left(2022-2019\right)$

$=\mathrm{1.2.3}+\mathrm{2.3.4}-\mathrm{1.2.3}+\mathrm{3.4.5}-\mathrm{2.3.4}+...+\mathrm{2020.2021.2022}-\mathrm{2019.2020.2021}$

$=\mathrm{2020.2021.2022}$

$\Rightarrow A=\frac{\mathrm{2020.2021.2022}}{3}$

2+6+12+20+30= 70

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

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

đưa ra dạng tổng quát giúp mk nhé thank you