x/1*2+x/2*3+x/3*4+x/4*5.+x/2012*2013=2012x(1/1*2+1/2*3+1/3*4+1/4*5.+1/2012*2013)=2012x(1-1/2+1/2-1/3+1/3-1/4+1/4-1/5.+1/2012-1/2013)=2012x(1-1/2013)=2012x*2012/2013=2012x/2013=1x=2013