已知1/x-1/x+1=1/x(x+1)求1/x(x+1)+1/(x+1)(x+2).1/(x+999)(x+1000)已知x/y=2求[2(x+y)-y(x+y)]/4x²-4xy+y²

Question

已知1/x-1/x+1=1/x(x+1)
求1/x(x+1)+1/(x+1)(x+2).1/(x+999)(x+1000)

已知x/y=2求[2(x+y)-y(x+y)]/4x²-4xy+y²

Answer 1

1/x(x+1)=(1/x-1/x+1)*1,
1/x(x+2)=(1/x-1/x+2)*1/2...
1/x(x+10)=(1/x-1/x+10)*1/10
(1)1/x(x+n)
=[1/x-1/(x+n)]*1/n
(2)1/(x+m)(x+n)
=[1/(x+m)-1/(x+n)]*1/(x+m-x-n)
=[1/(x+m)-1/(x+n)]*1/(m-n)
(3)若|ab-2|+(b-1)^2=0,求1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)+...+1(a+2011)(b+2011)+1/(a+2012)(b+2012)
|ab-2|+(b-1)^2=0,
则,ab-2=0且b-1=0
解得a=2,b=1
1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)+...+1(a+2011)(b+2011)+1/(a+2012)(b+2012)
=1/2+1/(3×2)+1/(4×3)+...+1/(2013×2012)+1/(2014×2013)
=1-1/2+1/2-1/3+1/3-1/4+...+1/2012-1/2013+1/2013-1/2014
=1-1/2014
=2013/2014