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2/1x2x3+2/2x3x4+2/3x4x5+⋯⋯+2/2...

162/54

2/(1x2x3)+2/(2x3x4)+2/(3x4x5)+.+2/(18x19x20) =(1/1x21/2x3)+(1/2x31/3x4)+(1/3x41/4x5)++(1/18x191/19x20) =1/1x21/19x20=189/380

解:原式=(1/1*2-1/2*3)+(1/2*3-1/3*4)+……(1/98*99-1/99*100)=1/1*2-1/99*100=49499/99000

1x2x3=1/4(1x2x3x4)2x3x4=1/4(2x3x4x5-1x2x3x4) ……7x8x9=1/4(7x8x9x10-6x7x8x9)1x2x3+2x3x4+3x4x5+.+7x8x9=1/4(7x8x9x10)=1260

这一般用拆项法,由公式:2/(k-1)k(k+1)=1/(k-1)-2/k+1/(k+1),得:原式=(1-2/2+1/3)+(1/2-2/3+1/4)+(1/3-2/4+1/5)++(1/97-2/98+1/99)+(1/98-2/99+1/100)=1-1/2-1/99+1/100=1/2-1/9900=4949/9900

2/1x2x3+2/2x3x4+2/3x4x5+.2/8x9x10= =2{1/2[1/1*2-1/2*3]+1/2[1/2*3-1/3*4]+1/2[1/3*4-1/4*5]++1/2[1/8*9-1/9*10] }=44/90=22/45

=1/1*2-1/2*3+1/2*3-1/3*4+1/3*4-1/4*5+1/2000*2001-1/2001*2002=1/1*2-1/2001*2002=1/2-1/4006002=100150/2003001

2/1x2x3+2/2x3x4+2/3x4x5+2/8x9x10= =2{1/2[1/1*2-1/2*3]+1/2[1/2*3-1/3*4]+1/2[1/3*4-1/4*5]++1/2[1/8*9-1/9*10] }=44/90=22/45

∵ 1x2x3+2x3x4++nx(n+1)x(n+2)= nx(n+1)x(n+2)x(n+3)÷ 4 ∴1x2x3+2x3x4++7x8x9= 7x(7+1)x(7+2)x(7+3)÷ 4=7x8x9x10÷ 4=1260

原式=(1/4)(1*2*3*4)+(1/4)(2*3*4*5-1*2*3*4)+(1/4)(3*4*5*6-2*3*4*5)+…… +(1/4)(7*8*9*10-6*7*8*9)=(1/4 )*( 7*8*9*10 - 1*2*3 *4)= (5040-24) *1/4=5016/4 =1254

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