\(5+\frac{1}{1+\frac{1}{1+\frac{2}{1+\frac{3}{4}}}}\) = \(5+\frac{1}{1+\frac{1}{1+\frac{2}{\frac{7}{4}}}}\)= \(5+\frac{1}{1+\frac{1}{1+\frac{8}{7}}}\) = \(5+\frac{1}{1+\frac{1}{\frac{15}{7}}}\) = \(5+\frac{1}{1+\frac{7}{15}}\) = \(5+\frac{1}{\frac{22}{15}}\)= \(5+\frac{15}{22}=\frac{125}{22}\)