Theo mình thì để tính nhanh được thì phân số cuối cùng phải là : $\dfrac{1}{14850}$
Đặt N = $\dfrac{1}{3}$ + $\dfrac{1}{9}$ + $\dfrac{1}{18}$ + $\dfrac{1}{30}$ + ..... + $\dfrac{1}{14850}$
=>$\dfrac{3}{2}$ N = $\dfrac{3}{2}$ . $\dfrac{1}{3}$ + $\dfrac{1}{9}$ + $\dfrac{1}{18}$ + $\dfrac{1}{30}$ + ..... + $\dfrac{1}{14850}$
=> $\dfrac{3}{2}$ N = $\dfrac{1}{2}$ + $\dfrac{1}{6}$ + $\dfrac{1}{12}$ + $\dfrac{1}{20}$ + .... + $\dfrac{1}{9900}$
=> $\dfrac{3}{2}$ N = $\dfrac{1}{1.2}$ + $\dfrac{1}{2.3}$ + $\dfrac{1}{3.4}$ + $\dfrac{1}{4.5}$ + .... + $\dfrac{1}{99.100}$
=> $\dfrac{3}{2}$ N = $\dfrac{1}{1}$ - $\dfrac{1}{2}$ + $\dfrac{1}{2}$ - $\dfrac{1}{3}$ + .... + $\dfrac{1}{99}$ - $\dfrac{1}{100}$
=> $\dfrac{3}{2}$ N = $\dfrac{1}{1}$ - $\dfrac{1}{100}$
=> $\dfrac{3}{2}$ N = $\dfrac{99}{100}$
=> N = $\dfrac{99}{100}$ : $\dfrac{3}{2}$
=> N = $\dfrac{33}{50}$
