题目

已知1-1/2=1/2,1/2-1/3=1/6,1/3-1/4=1/12,...根据这些等式解答下列各题.
(1)求值:1/1*2+1/2*3+1/3*4+1/4*5+1/5*6
(2)化简:1/1*2+1/2*3+1/3*4+...+1/n(n+1)
(3)若1/1*2+1/2*3+1/3*4+...+1/n(n+1)=19/20,则n=?

提问时间：2020-11-03

答案

1/1*2+1/2*3+1/3*4+1/4*5+1/5*6
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)
=1-1/6
=5/6
1/1*2+1/2*3+1/3*4+...+1/n(n+1)
=(1-1/2)+(1/2-1/3)+……+[1/n-1/(n+1)
=1-1/(n+1)
=n/(n+1)
1/1*2+1/2*3+1/3*4+...+1/n(n+1)=19/20
所以n/(n+1)=19/(19+1)
所以n=19