Ta có
$1212 = 12.101, 1414 = 14.101,..., 9898 = 98.101$
Vậy ta có
$S = \dfrac{505}{10.1212} + \dfrac{505}{12.1414} + \cdots + \dfrac{505}{96.9898}$
$= \dfrac{5.101}{10.12.101} + \dfrac{5.101}{12 . 14 . 101} + \cdots + \dfrac{5.101}{96.98.101}$
$= \dfrac{5}{10.12} + \dfrac{5}{12.14} + \cdots + \dfrac{5}{96.98}$
Nhân 2 vào 2 vế ta có
$S = 5 . \dfrac{2}{10.12} + 5.\dfrac{2}{12.14} + \cdots + 5 . \dfrac{2}{96.98}$
$= 5 \left( \dfrac{1}{10.12} + \dfrac{1}{12.14} + \cdots + \dfrac{1}{96.98} \right)$
$= 5\left( \dfrac{1}{10} - \dfrac{1}{12} + \dfrac{1}{12} - \dfrac{1}{14} + \cdots + \dfrac{1}{96} - \dfrac{1}{98} \right)$
$= 5 \left( \dfrac{1}{10} - \dfrac{1}{98} \right)$
$= 5 . \dfrac{22}{245} = \dfrac{22}{49}$
Vậy $S = \dfrac{22}{49}$
