利用等差数列公式.
x+x/(1+2)+x/(1+2+3)+.+x/(1+2+...+2009)
=x/(1*2/2)+x/(2*3/2)+...+x/(2009*2010/2) (等差数列公式)
=2x/(1*2)+2x/(2*3)+...+2x/(2009*2010) (把分母中的除号倒上去)
=2x(1/(1*2)+1/(2*3)+1/(3*4)+...+1/(2009*2010)) (提取公因式)
=2x((1/1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/2009-1/2010)) (裂项法)
=2x(1/1-1/2+1/2-1/3+1/3-1/4+...-1/2010)) (去括号)
=2x(1/1+(-1/2+1/2)+(-1/3+1/3)+...+(-1/2009+1/2009)-1/2010) (添括号)
=2x(1/1-1/2010)
=2x(1-1/2010)
=2x * 2009/2010
=2009/1005 * x
得2009/1005 * x = 2009 （题目）
x = 1005