Question

[công chúa mặt trăng]

 Tính tích     S =  1x2 + 2x3 + 3x4 + ……………… + 11x12 + 12x13

Answer 1

[Trần Hà Quỳnh Như]

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 

Answer 2

[Yêu Toán]

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=\(\frac{12\times13\times14}{3}=728\)