x/(1×2)+x/(2×3)+x/(3×4)...+x/(2009×2010)=2009
x[1/(1×2)+1/(2×3)+1/(3×4)+...+1/(2009×2010)]=2009
x/(1-1/2+1/2-1/3+1/3-1/4+...+1/2009-1/2010)=2009
x/(1-1/2010)=2009
2009x/2010=2009
x=2010
x/(1×2)+x/(2×3)+x/(3×4)...+x/(2009×2010)=2009
x[1/(1×2)+1/(2×3)+1/(3×4)+...+1/(2009×2010)]=2009
x/(1-1/2+1/2-1/3+1/3-1/4+...+1/2009-1/2010)=2009
x/(1-1/2010)=2009
2009x/2010=2009
x=2010