$\text{Áp dụng công thức tính tổng: 1+2+3+4+5+...+n=n×( n+1)÷2}$
$\text{Ta có A = $\frac{1}{2 × (2 + 1) ÷2}$ + $\frac{1}{3 × (3 + 1) ÷2}$ + $\frac{1}{4 × (4 + 1) ÷2}$ + ... + $\frac{1}{2009 × (2009 + 1) ÷2}$}$
$\text{A = $\frac{2}{2 × 3}$ + $\frac{2}{3 × 4}$ + $\frac{2}{4 × 5}$ + ... + $\frac{2}{2009 × 2010}$}$
$\text{A = 2 × ($\frac{1}{2 × 3}$ + $\frac{1}{3 × 4}$ + $\frac{1}{4 × 5}$ + ... + $\frac{1}{2009 × 2010}$)}$
$\text{A = 2 × ($\frac{1}{2}$ - $\frac{1}{3}$ + $\frac{1}{4}$ - $\frac{1}{5}$ + $\frac{1}{6}$ - $\frac{1}{7}$ + ....+ $\frac{1}{2009}$ - $\frac{1}{2010}$)}$
$\text{A = 2 × ($\frac{1}{2}$ - $\frac{1}{2010}$)}$
$\text{A = $\frac{1004}{1005}$}$
