Solve 2/1+2{1+frac{2/1+frac{2{1+2}}}}

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

Evaluate

$$\frac{22}{21}\approx 1.047619048$$

Solution Steps

Add 1 and 2 to get 3.

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

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

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

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

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

Add 3 and 2 to get 5.

$$\frac{2}{1+\frac{2}{1+\frac{2}{\frac{5}{3}}}}$$

Divide 2 by \frac{5}{3} by multiplying 2 by the reciprocal of \frac{5}{3}.

$$\frac{2}{1+\frac{2}{1+2\times \frac{3}{5}}}$$

Express 2\times \frac{3}{5} as a single fraction.

$$\frac{2}{1+\frac{2}{1+\frac{2\times 3}{5}}}$$

Multiply 2 and 3 to get 6.

$$\frac{2}{1+\frac{2}{1+\frac{6}{5}}}$$

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

$$\frac{2}{1+\frac{2}{\frac{5}{5}+\frac{6}{5}}}$$

Since \frac{5}{5} and \frac{6}{5} have the same denominator, add them by adding their numerators.

$$\frac{2}{1+\frac{2}{\frac{5+6}{5}}}$$

Add 5 and 6 to get 11.

$$\frac{2}{1+\frac{2}{\frac{11}{5}}}$$

Divide 2 by \frac{11}{5} by multiplying 2 by the reciprocal of \frac{11}{5}.

$$\frac{2}{1+2\times \frac{5}{11}}$$

Express 2\times \frac{5}{11} as a single fraction.

$$\frac{2}{1+\frac{2\times 5}{11}}$$

Multiply 2 and 5 to get 10.

$$\frac{2}{1+\frac{10}{11}}$$

Convert 1 to fraction \frac{11}{11}.

$$\frac{2}{\frac{11}{11}+\frac{10}{11}}$$

Since \frac{11}{11} and \frac{10}{11} have the same denominator, add them by adding their numerators.

$$\frac{2}{\frac{11+10}{11}}$$

Add 11 and 10 to get 21.

$$\frac{2}{\frac{21}{11}}$$

Divide 2 by \frac{21}{11} by multiplying 2 by the reciprocal of \frac{21}{11}.

$$2\times \frac{11}{21}$$

Express 2\times \frac{11}{21} as a single fraction.

$$\frac{2\times 11}{21}$$

Multiply 2 and 11 to get 22.

$$\frac{22}{21}$$

Factor

$$\frac{2 \cdot 11}{3 \cdot 7} = 1\frac{1}{21} = 1.0476190476190477$$