Question

观察 1/1*2 + 1/2*3 =(1-1/2)(1/2 - 1/3)=1-1/3=2/3,1/1*2 + 1/2*3

Answer 1

1.数列的规律是：1/1*2 + 1/2*3 + 1/3*4 +……+1/n(n+1)=1 - 1/n+1 =n/(n+1)

1/1*2 + 1/2*3 + 1/3*4+…+1/99*100=1 -1/100=99/100

2.原式=(1/2)[(1/2)-(1/4)+(1/4)-(1/6)+(1/6)-(1/8)+.+(1/2n)- 1/(2n+2)]=1001/4008

(1/2)[(1/2) - 1/(2n+2)]=1001/4008

(1/2) - 1/(2n+2)=1001/2004

2n+2=2004

n=1001

3.1/m(m+1) =1/m -1/(m+1)