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 

S = 728

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 

ai k minh minh k lai cho

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 

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

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

S = 1.2+2.3+3.4+....+38.39+39.40

S.3 = ( 1.2+2.3+3.4+...+38.39+39.40 ) . 3

S.3 = 1.2.(3-0)+2.3.(4-1)+3.4.(5-2)+...........+38.39.(40-37)+39.40.(41-38)

S.3 = 1.2.3-0.1.2+2.3.4-1.2.3+3.4.5-2.3.4+...........+38.39.40-37.38.39+39.40.41-38.39.40

S = \(\frac{39.40.41}{3}\)

S = 13.40.41

S = 21320

S= 1.2 + 2.3 + ... + 39.40

=> 3S = 3.(1.2 + 2.3 + ... + 39.40)

=> 3S = 1.2.(3-0) + 2.3.(4-1) +....+39.40.(41-38)

=> 3S = 1.2.3 - 0.1.2 + 2.3.4 - 1.2.3 + ... + 39.40.41 - 38.39.40

=> 3S = 39.40.41

=> S = \(\frac{39.40.41}{3}\)

=> S=21320

S=1.2+2.3+3.4+.....+38.39+39.40

=>3S=1.2.3+2.3.3+3.3.4+...+3.38.39+39.40.3

=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+38.39.40-37.38.39+39.40.41-38.39.40

=39.40.41

=>S=13.40.41

S = 1 x 2 + 2 x 3 = 3 x 4 + ... + 38 x 39 + 39 x 40

3S = 1 x 2 x 3 + 2 x 3 x 3 + 3 x 4 x 3+ ... + 38 x 39 x 3 + 39 x 40 x 3

3S = 1 x 2 x 3 + 2 x 3 x ( 4 - 1 ) + 3 x 4 x ( 5 - 2 ) + ... + 38 x 39 x ( 40 - 37 ) + 39 x 40 x ( 41 - 38 )

3S = 1 x 2 x 3 + 2 x 3 x 4 - 1 x 2 x 3 + 3 x 4 x 5 - 2 x 3 x 4 + ... + 38 x 39 x 40 - 37 x 38 x 39 + 39 x 40 x 41 - 38 x 39 x 40

 S = 39 x 40 x 41 : 3 

 S = 21320

Kết quả là 3333

♣ Còn phần công thức, bạn có thể xem ở link sau bài 3 : https://sites.google.com/site/toantieuhocpl/20-tinh-nhanh 

Gọi biểu thức trên là S, ta có :

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

100S x 3 = 1x2x3 + 2x3x3 + 3x4x3 + 4x5x3 + ... + 99x100x3

100S x 3 = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + 4x5x(6-3) + ... + 99x100x(101-98)

100S x 3 = 1x2x3 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + 4x5x6 - 3x4x5 + ... + 99x100x101 - 98x99x100.

100S x 3 = 99x100x101

100S = 99x100x101 : 3

100S = 333300

S=333300:100

S=3333

 S = 1 x 2 + 2 x 3 +3 x 4 + 4 x 5 + .... + 28 x 29 + 29 x 30 
S = [29x(29+1)x(29+2)]/3 
S = 8990