Solve 1+1/1+1{1+1/1+1{1+frac{1/1+frac{1{1+frac{1/1+frac{1{1+frac{1}{1+frac{1}{1}}}}}}1111}}}}

$$1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1 } } } } } } 1111 } } } }$$

Evaluate

$$\frac{44544}{26729}\approx 1.666504546$$

Solution Steps

Divide 1 by 1 to get 1.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+1}}}}}\times 1111}}}}$$

Add 1 and 1 to get 2.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{2}}}}}\times 1111}}}}$$

Convert 1 to fraction \frac{2}{2}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{2}{2}+\frac{1}{2}}}}}\times 1111}}}}$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{2+1}{2}}}}}\times 1111}}}}$$

Add 2 and 1 to get 3.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{3}{2}}}}}\times 1111}}}}$$

Divide 1 by \frac{3}{2} by multiplying 1 by the reciprocal of \frac{3}{2}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+1\times \frac{2}{3}}}}\times 1111}}}}$$

Multiply 1 and \frac{2}{3} to get \frac{2}{3}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{2}{3}}}}\times 1111}}}}$$

Convert 1 to fraction \frac{3}{3}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{3}{3}+\frac{2}{3}}}}\times 1111}}}}$$

Since \frac{3}{3} and \frac{2}{3} have the same denominator, add them by adding their numerators.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{3+2}{3}}}}\times 1111}}}}$$

Add 3 and 2 to get 5.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{5}{3}}}}\times 1111}}}}$$

Divide 1 by \frac{5}{3} by multiplying 1 by the reciprocal of \frac{5}{3}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+1\times \frac{3}{5}}}\times 1111}}}}$$

Multiply 1 and \frac{3}{5} to get \frac{3}{5}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{3}{5}}}\times 1111}}}}$$

Convert 1 to fraction \frac{5}{5}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{5}{5}+\frac{3}{5}}}\times 1111}}}}$$

Since \frac{5}{5} and \frac{3}{5} have the same denominator, add them by adding their numerators.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{5+3}{5}}}\times 1111}}}}$$

Add 5 and 3 to get 8.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{8}{5}}}\times 1111}}}}$$

Divide 1 by \frac{8}{5} by multiplying 1 by the reciprocal of \frac{8}{5}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+1\times \frac{5}{8}}\times 1111}}}}$$

Multiply 1 and \frac{5}{8} to get \frac{5}{8}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{5}{8}}\times 1111}}}}$$

Convert 1 to fraction \frac{8}{8}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{8}{8}+\frac{5}{8}}\times 1111}}}}$$

Since \frac{8}{8} and \frac{5}{8} have the same denominator, add them by adding their numerators.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{8+5}{8}}\times 1111}}}}$$

Add 8 and 5 to get 13.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{13}{8}}\times 1111}}}}$$

Divide 1 by \frac{13}{8} by multiplying 1 by the reciprocal of \frac{13}{8}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+1\times \frac{8}{13}\times 1111}}}}$$

Multiply 1 and \frac{8}{13} to get \frac{8}{13}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{8}{13}\times 1111}}}}$$

Express \frac{8}{13}\times 1111 as a single fraction.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{8\times 1111}{13}}}}}$$

Multiply 8 and 1111 to get 8888.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{8888}{13}}}}}$$

Convert 1 to fraction \frac{13}{13}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{13}{13}+\frac{8888}{13}}}}}$$

Since \frac{13}{13} and \frac{8888}{13} have the same denominator, add them by adding their numerators.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{13+8888}{13}}}}}$$

Add 13 and 8888 to get 8901.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{8901}{13}}}}}$$

Divide 1 by \frac{8901}{13} by multiplying 1 by the reciprocal of \frac{8901}{13}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+1\times \frac{13}{8901}}}}$$

Multiply 1 and \frac{13}{8901} to get \frac{13}{8901}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{13}{8901}}}}$$

Convert 1 to fraction \frac{8901}{8901}.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{8901}{8901}+\frac{13}{8901}}}}$$

Since \frac{8901}{8901} and \frac{13}{8901} have the same denominator, add them by adding their numerators.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{8901+13}{8901}}}}$$

Add 8901 and 13 to get 8914.

$$1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{8914}{8901}}}}$$

Divide 1 by \frac{8914}{8901} by multiplying 1 by the reciprocal of \frac{8914}{8901}.

$$1+\frac{1}{1+\frac{1}{1+1\times \frac{8901}{8914}}}$$

Multiply 1 and \frac{8901}{8914} to get \frac{8901}{8914}.

$$1+\frac{1}{1+\frac{1}{1+\frac{8901}{8914}}}$$

Convert 1 to fraction \frac{8914}{8914}.

$$1+\frac{1}{1+\frac{1}{\frac{8914}{8914}+\frac{8901}{8914}}}$$

Since \frac{8914}{8914} and \frac{8901}{8914} have the same denominator, add them by adding their numerators.

$$1+\frac{1}{1+\frac{1}{\frac{8914+8901}{8914}}}$$

Add 8914 and 8901 to get 17815.

$$1+\frac{1}{1+\frac{1}{\frac{17815}{8914}}}$$

Divide 1 by \frac{17815}{8914} by multiplying 1 by the reciprocal of \frac{17815}{8914}.

$$1+\frac{1}{1+1\times \frac{8914}{17815}}$$

Multiply 1 and \frac{8914}{17815} to get \frac{8914}{17815}.

$$1+\frac{1}{1+\frac{8914}{17815}}$$

Convert 1 to fraction \frac{17815}{17815}.

$$1+\frac{1}{\frac{17815}{17815}+\frac{8914}{17815}}$$

Since \frac{17815}{17815} and \frac{8914}{17815} have the same denominator, add them by adding their numerators.

$$1+\frac{1}{\frac{17815+8914}{17815}}$$

Add 17815 and 8914 to get 26729.

$$1+\frac{1}{\frac{26729}{17815}}$$

Divide 1 by \frac{26729}{17815} by multiplying 1 by the reciprocal of \frac{26729}{17815}.

$$1+1\times \frac{17815}{26729}$$

Multiply 1 and \frac{17815}{26729} to get \frac{17815}{26729}.

$$1+\frac{17815}{26729}$$

Convert 1 to fraction \frac{26729}{26729}.

$$\frac{26729}{26729}+\frac{17815}{26729}$$

Since \frac{26729}{26729} and \frac{17815}{26729} have the same denominator, add them by adding their numerators.

$$\frac{26729+17815}{26729}$$

Add 26729 and 17815 to get 44544.

$$\frac{44544}{26729}$$

Factor

$$\frac{2 ^ {9} \cdot 3 \cdot 29}{26729} = 1\frac{17815}{26729} = 1.6665045456246026$$