Giải thích các bước giải:
Ta có:
$A=\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{x(x+2)}$
$\to A=\dfrac{3-1}{1.3}+\dfrac{5-3}{3.5}+\dfrac{7-5}{5.7}+...+\dfrac{x+2-x}{x(x+2)}$
$\to A=\dfrac11-\dfrac13+\dfrac13-\dfrac15+\dfrac15-\dfrac17+...+\dfrac1{x}-\dfrac1{x+2}$
$\to A=1-\dfrac1{x+2}$
Để $A<\dfrac{2003}{2004}$
$\to 1-\dfrac1{x+2}<\dfrac{2003}{2004}$
$\to \dfrac1{x+2}>1-\dfrac{2003}{2004}$
$\to \dfrac1{x+2}>\dfrac1{2004}$
$\to x+2<2004$
$\to x<2002$
Mà $x\in N^*\to x\in\{1, 2, 3, 4, .., 2001\}$
