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

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

Evaluate

$$\frac{19}{7}\approx 2.714285714$$

Solution Steps

Add 2 and 2 to get 4.

$$2+\frac{2}{2+\frac{2}{2+\frac{2}{4}}}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

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

Convert 2 to fraction \frac{4}{2}.

$$2+\frac{2}{2+\frac{2}{\frac{4}{2}+\frac{1}{2}}}$$

Since \frac{4}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$2+\frac{2}{2+\frac{2}{\frac{4+1}{2}}}$$

Add 4 and 1 to get 5.

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

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

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

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

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

Multiply 2 and 2 to get 4.

$$2+\frac{2}{2+\frac{4}{5}}$$

Convert 2 to fraction \frac{10}{5}.

$$2+\frac{2}{\frac{10}{5}+\frac{4}{5}}$$

Since \frac{10}{5} and \frac{4}{5} have the same denominator, add them by adding their numerators.

$$2+\frac{2}{\frac{10+4}{5}}$$

Add 10 and 4 to get 14.

$$2+\frac{2}{\frac{14}{5}}$$

Divide 2 by \frac{14}{5} by multiplying 2 by the reciprocal of \frac{14}{5}.

$$2+2\times \frac{5}{14}$$

Express 2\times \frac{5}{14} as a single fraction.

$$2+\frac{2\times 5}{14}$$

Multiply 2 and 5 to get 10.

$$2+\frac{10}{14}$$

Reduce the fraction \frac{10}{14} to lowest terms by extracting and canceling out 2.

$$2+\frac{5}{7}$$

Convert 2 to fraction \frac{14}{7}.

$$\frac{14}{7}+\frac{5}{7}$$

Since \frac{14}{7} and \frac{5}{7} have the same denominator, add them by adding their numerators.

$$\frac{14+5}{7}$$

Add 14 and 5 to get 19.

$$\frac{19}{7}$$

Factor

$$\frac{19}{7} = 2\frac{5}{7} = 2.7142857142857144$$