1/4*5*6+1/5*6*7+1/6*7*8+1/7*8*9+......1/98*99*100

解：1/(4×5×6)=1/2[1/(4×5)-1/(5×6)]
1/(5×6×7)=1/2[1/(5×6)- 1/(6×7 )]
1/(6×7×8)=1/2[1/(6×7)- 1/(7×8)]
………
1/(98×99×100)=1/2[1/(98×99)-1/(99×100)]
∴1/(4×5×6)+ 1/(5×6×7)+ 1/(6×7×8)+…..+ 1/(98×99×100)
=1/2[1/(4×5)-1/(5×6)]+ 1/2[1/(5×6)- 1/(6×7 )]+ 1/2[1/(6×7)- 1/(7×8)]+ 1/2[1/(98×99)-1/(99×100)]
=1/2[1/(4×5)-1/(5×6)+ 1/(5×6)- 1/(6×7 )+ 1/(6×7)- 1/(7×8)+…..+ 1/(98×99)-1/(99×100)]
=1/2[1/(4×5) -1/(99×100)]
=1/2×494/9900
=247/9900本回答被提问者采纳

1/[(n+3)(n+4)(n+5)]=(1/2){1/[(n+3)(n+4)] - 1/[(n+4)(n+5)]}
1/[4*5*6]+1/[5*6*7]+...+1/[98*99*100]
=(1/2){1/[4*5]-1/[5*6]+1/[5*6]-1/[6*7]+...+1/[98*99]-1/[99*100]}
=(1/2){1/[4*5]-1/[99*100]}