Discussion

Explanation:

$\frac{1}{1+{\displaystyle \frac{1}{3-{\displaystyle \frac{4}{2+{\displaystyle \frac{1}{3-{\displaystyle \frac{1}{2}}}}}}}}}+\frac{3}{3-{\displaystyle \frac{4}{3+{\displaystyle \frac{1}{2-{\displaystyle \frac{1}{2}}}}}}}$

$=\frac{1}{1+{\displaystyle \frac{1}{3-{\displaystyle \frac{4}{2+{\displaystyle \frac{2}{5}}}}}}}+\frac{3}{3-{\displaystyle \frac{4}{3+{\displaystyle \frac{1}{{\displaystyle \frac{3}{2}}}}}}}$

$=\frac{1}{1+{\displaystyle \frac{1}{3-{\displaystyle \frac{4}{{\displaystyle \frac{12}{5}}}}}}}+\frac{3}{3-{\displaystyle \frac{4}{3+{\displaystyle \frac{1}{{\displaystyle \frac{3}{2}}}}}}}$

$=\frac{1}{1+{\displaystyle \frac{1}{3-{\displaystyle \frac{5}{3}}}}}+\frac{3}{3-{\displaystyle \frac{4}{3+{\displaystyle \frac{2}{3}}}}}=\frac{1}{1+{\displaystyle \frac{1}{{\displaystyle \frac{4}{3}}}}}+\frac{3}{3-{\displaystyle \frac{4}{{\displaystyle \frac{11}{3}}}}}$

$=\frac{1}{1+{\displaystyle \frac{3}{4}}}+\frac{3}{3-{\displaystyle \frac{12}{11}}}=\frac{1}{3-{\displaystyle \frac{12}{11}}}+\frac{3}{3-{\displaystyle \frac{12}{11}}}=\frac{1}{{\displaystyle \frac{7}{4}}}+\frac{3}{{\displaystyle \frac{21}{11}}}$

$=\frac{4}{7}+\frac{11}{7}=\frac{15}{7}$

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