1/1*2*3+1/2*3*4+…+1/98*99*100

1.公式：1/[n*(n+1)*(n+2)]=1/2*{1/[n*(n+1)]-1/[(n+1)*(n+2)]}
原式=1/2*[1/(1*2)-1/(2*3)+1/(2*3)-1/(3*4)+...+1/(98*99)-1/(99*100)]
=1/2*[1/2-1/9900]
=1/2 * 4949/9900
=4949/19800
2.原式=3/4*8/9*15/16*24/25*.*2499/2500=(3*8*15*24*35*.*2499)/(4*9*16*25*36*...*2500)=(3*2*4*3*5*4*6*5*7*6*8*...*49*51)/(2*2*3*3*4*4*5*5*6*6*.*50*50)=51/100
3.公式：1/n+(n+1)+(n+2)=2/(n+2)*(n+3)
原式=1+2*[1/(2*3)+1/(3*4)+...+1/(100*101)
=1+2*[1/2 - 1/3 + 1/3 - 1/4 + ...+ 1/100 -1/101]
=1+2*[1/2 - 1/101]
=1+2* 100/202
=1+ 100/101
=201/101