1/1*2+1/2*3+1/3*4+.+1/1000*1001求简便算式为什么，

1/1*2+1/2*3+1/3*4+......+1/1000*1001
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+……+（1/1000-1/1001）
=1-1/1001
=1000/1001

公式 1/M*N=1/M-1/N (M1/1*2+1/2*3+1/3*4+......+1/1000*1001=1-1/2+1/2-1/3+1/3-1/4…………+1/1000-1/1001=1-1/1001=1000/1001

1/[N*(N+1)]=1/N-1/(N+1)
所以，
原式
=（1-1/2）+(1/2-1/3)+(1/3-1/4)+……+[1/N-1/(N+1)]
=1-1/(N+1)=N/(N+1)
=1000/1001

1/1*2+1/2*3+1/3*4+......+1/1000*1001
=1-1/2+1/2-1/3+...+1/1000-1/1001
=1-1/1001
=1000/1001

1/n(n+1)=1/n-1/(n+1)
原式=(1-1/2)+(1/2-1/3)+……+(1/1000-1/1001)
=1-1/1001
=1000/1001

1-1/1001=1000/1001

1/1*2+1/2*3+1/3*4+......+1/1000*1001
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+......+(1/1000-1/1001)]
=1-1/1001
=1000/1001