$1+\frac{1}{2}=\frac{3}{2}$ $1+\frac{1}{\frac{3}{2}}=1+\frac{2}{3}=\frac{5}{3}$ $1+\frac{1}{\frac{5}{3}}=1+\frac{3}{5}=\frac{8}{5}$ $1+\frac{1}{\frac{8}{5}}=1+\frac{5}{8}=\frac{13}{8}$ $\frac{1}{\frac{13}{8}}=\frac{8}{13}$