$\ \dfrac{191919}{121212} + \dfrac{191919}{202020} + \dfrac{191919}{303030} + \dfrac{191919}{424242} + \dfrac{191919}{565656}$
$\ = \dfrac{191919 : 10101}{121212 : 10101} + \dfrac{191919 : 10101}{202020 : 10101} + \dfrac{191919 : 10101}{303030 : 10101} + \dfrac{191919 : 10101}{424242 : 10101} + \dfrac{191919 : 10101}{565656 : 10101}$
$\ = \dfrac{19}{12} + \dfrac{19}{20} + \dfrac{19}{30} + \dfrac{19}{42} + \dfrac{19}{56}$
$\ = 19(\dfrac{1}{3.4} + \dfrac{1}{4.5} + \dfrac{1}{5.6} + \dfrac{1}{6.7} + \dfrac{1}{7.8})$
$\ = 19(\dfrac{1}{3} - \dfrac{1}{4} + \dfrac{1}{4} - \dfrac{1}{5} + \dfrac{1}{5} - \dfrac{1}{6} + \dfrac{1}{6} - \dfrac{1}{7} + \dfrac{1}{7} - \dfrac{1}{8})$
$\ = 19(\dfrac{1}{3} - \dfrac{1}{8})$
$\ = 19. \dfrac{8-3}{24} = 19 . \dfrac{5}{24} = 3\dfrac{23}{24}$
