1x2x3+2x3x4+3x4x5+.+10x11x12

来源：学生作业帮助网编辑：作业帮时间：2024/05/09 15:30:59

1x2x3+2x3x4+3x4x5+.+10x11x12
1x2x3+2x3x4+3x4x5+.+10x11x12

1x2x3+2x3x4+3x4x5+.+10x11x12
设第n项为an
an=n(n+1)(n+2)=n^3+3n^2+2n
1×2×3+2×3×4+...+10×11×12
=(1^3+2^3+...+10^3)+3(1^2+2^2+...+10^2)+2(1+2+...+10)
=[10(10+1)/2]^2+3×10×(10+1)(20+1)/6+2×10×11/2
=3025+1155+110
=4290
用到的公式：
1^3+2^3+...+n^3=[n(n+1)/2]^2
1^2+2^2+...+n^2=n(n+1)(2n+1)/6
1+2+3+...+n=n(n+1)/2

1×2×3＋2×3×4＋3×4×5＋……＋10×11×12
＝10×(10＋1)×(10＋2)×(10＋3)/4
＝4290

∑▒〖k(k+10(k+2)〗=∑▒〖(k^3+3k^2+2k)〗，∑_1^n▒n^3 =[n(n+1)/2]^2，∑_1^n▒n^2 =n(n+1)(2n+1)/6，∑_1^n▒n=n(n+1)/2
所以原式=∑_1^10▒〖(k^3+3k^2+2k)〗=[(10*(10+1))/2]^2+3*(10*(10+1)*(2*10+1))/6+10*(10+1)=4290

1x2x3+2x3x4+3x4x5+…+8x9x10 1x2x3+2x3x4+3x4x5+.+10x11x12 1x2x3+2x3x4+3x4x5+...+7x8x9=, 1x2X3+2x3X4+3x4X5+…+7X8X9=? 1x2x3 2x3x4 3x4x5 .10x11x12,怎么算求和：1X2X3+2X3X4+3X4X5+...+98X99X100 1x2=1|3(1x2x3-0x1x2) 2x3=1|3(2x3x4-1x2x3) 3x4=1|3(3x4x5-2x3x4)由此推断1x2x3+2x3x4+3x4x5.7x8x9=? 1x2=1/3x(1x2x3-0x1x2)2x3=1/3x(2x3x4-1x2x3)3x4=1/3x(3x4x5-2x3x4)求1x2x3+2x3x4+3x4x5+...+7x8x9=? 1/1x2x3+1/2x3x4+1/3x4x5+.+1/n(n+1)(n+2) 2/1x2x3+2/2x3x4+2/3x4x5+2/4x5x6+...+2/98x99x100= 1/1x2x3+1/2x3x4+1/3x4x5+.+1/8x9x10 求简算请1/1x2x3+1/2x3x4+1/3x4x5+1/4x5x6= 1/1x2x3+1/2x3x4+1/3x4x5+------+1/98x99x100= 1/(1x2x3)+1/(2x3x4)+1/(3x4x5)+.1/(20x21x22)=? 1/1x2x3+1/2x3x4+1/3x4x5+.+1/9x10x11= 1/1x2x3+1/2x3x4+1/3x4x5+.1/98x99x100=要有中间过程 1/1x2x3+1/2x3x4+1/3x4x5+.+1/11x12x13= 1x2x3+2x3x4+3x4x5+4x5x6+...+n(n+1)(n+2)=