Questions mathématiques diverses

Question 57:
      

Réponse 57:
$1+\dfrac{1+\dfrac{1+\dfrac{1+\dfrac{\dfrac{3}{2}}{2}}{2}}{2}}{1+\dfrac{2}{1+\dfrac{2}{1+\dfrac{2}{3}}}}$ $=1+\dfrac{1+\dfrac{1+\dfrac{1+\dfrac{3}{4}}{2}}{2}}{1+\dfrac{2}{1+\dfrac{2}{1+\dfrac{2}{3}}}}$ $=1+\dfrac{1+\dfrac{1+\dfrac{\dfrac{7}{4}}{2}}{2}}{1+\dfrac{2}{1+\dfrac{2}{\dfrac{5}{3}}}}$ $=1+\dfrac{1+\dfrac{1+\dfrac{7}{8}}{2}}{1+\dfrac{2}{1+\dfrac{6}{5}}}$ $=1+\dfrac{1+\dfrac{\dfrac{15}{8}}{2}}{1+\dfrac{2}{\dfrac{11}{5}}}$ $=1+\dfrac{1+\dfrac{15}{16}}{1+\dfrac{10}{11}}$ $=1+\dfrac{\dfrac{31}{16}}{\dfrac{21}{11}}$ $=1+\dfrac{31}{16}\times\dfrac{11}{21}$ $=1+\dfrac{341}{336}=\dfrac{677}{336}$