Question

$\frac{x}{2}+ \frac{x}{6} + \frac{x}{12} +...+ \frac{x}{2009*2010}+ \frac{x}{2010*2011} =2010$

Answer 1

x/2+x/6+x/12+.....x/2009*2010+x/2010*2011=2010
x(1/2+1/6+1/12+.....1/2009*2010+1/2010*2011)=2010
x × 2010/2011 = 2010
x = 2011

1/2=1-1/2
1/6=1/2-1/3
1/12=1/3-1/4
..........
1/2009*2010=1/2009-1/2010
1/2010*2011=1/2010-1/2011
1/2+1/6+1/12+.....1/2009*2010+1/2010*2011=(1-1/2)+(1/2-1/3)+(1/3-1/4)+.........+  +(1/2009-1/2010)+(1/2010-1/2011)=1-1/2011=(2011-1)/2011 = 2010/2011