Solve 3+1/7+1{15+1/1+1{288+frac{1/1+frac{1{2+frac{1/1+frac{1{3}}}}}}}}

$$3+ \frac{ 1 }{ 7+ \frac{ 1 }{ 15+ \frac{ 1 }{ 1+ \frac{ 1 }{ 288+ \frac{ 1 }{ 1+ \frac{ 1 }{ 2+ \frac{ 1 }{ 1+ \frac{ 1 }{ 3 } } } } } } } }$$

Evaluate

$$\frac{1542500}{490993}\approx 3.14159265$$

Solution Steps

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

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{1+\frac{1}{2+\frac{1}{\frac{3}{3}+\frac{1}{3}}}}}}}}$$

Since \frac{3}{3} and \frac{1}{3} have the same denominator, add them by adding their numerators.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{1+\frac{1}{2+\frac{1}{\frac{3+1}{3}}}}}}}}$$

Add 3 and 1 to get 4.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{1+\frac{1}{2+\frac{1}{\frac{4}{3}}}}}}}}$$

Divide 1 by \frac{4}{3} by multiplying 1 by the reciprocal of \frac{4}{3}.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{1+\frac{1}{2+1\times \frac{3}{4}}}}}}}$$

Multiply 1 and \frac{3}{4} to get \frac{3}{4}.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{1+\frac{1}{2+\frac{3}{4}}}}}}}$$

Convert 2 to fraction \frac{8}{4}.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{1+\frac{1}{\frac{8}{4}+\frac{3}{4}}}}}}}$$

Since \frac{8}{4} and \frac{3}{4} have the same denominator, add them by adding their numerators.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{1+\frac{1}{\frac{8+3}{4}}}}}}}$$

Add 8 and 3 to get 11.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{1+\frac{1}{\frac{11}{4}}}}}}}$$

Divide 1 by \frac{11}{4} by multiplying 1 by the reciprocal of \frac{11}{4}.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{1+1\times \frac{4}{11}}}}}}$$

Multiply 1 and \frac{4}{11} to get \frac{4}{11}.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{1+\frac{4}{11}}}}}}$$

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

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{\frac{11}{11}+\frac{4}{11}}}}}}$$

Since \frac{11}{11} and \frac{4}{11} have the same denominator, add them by adding their numerators.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{\frac{11+4}{11}}}}}}$$

Add 11 and 4 to get 15.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{1}{\frac{15}{11}}}}}}$$

Divide 1 by \frac{15}{11} by multiplying 1 by the reciprocal of \frac{15}{11}.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+1\times \frac{11}{15}}}}}$$

Multiply 1 and \frac{11}{15} to get \frac{11}{15}.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{288+\frac{11}{15}}}}}$$

Convert 288 to fraction \frac{4320}{15}.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{\frac{4320}{15}+\frac{11}{15}}}}}$$

Since \frac{4320}{15} and \frac{11}{15} have the same denominator, add them by adding their numerators.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{\frac{4320+11}{15}}}}}$$

Add 4320 and 11 to get 4331.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{\frac{4331}{15}}}}}$$

Divide 1 by \frac{4331}{15} by multiplying 1 by the reciprocal of \frac{4331}{15}.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+1\times \frac{15}{4331}}}}$$

Multiply 1 and \frac{15}{4331} to get \frac{15}{4331}.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{15}{4331}}}}$$

Convert 1 to fraction \frac{4331}{4331}.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{\frac{4331}{4331}+\frac{15}{4331}}}}$$

Since \frac{4331}{4331} and \frac{15}{4331} have the same denominator, add them by adding their numerators.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{\frac{4331+15}{4331}}}}$$

Add 4331 and 15 to get 4346.

$$3+\frac{1}{7+\frac{1}{15+\frac{1}{\frac{4346}{4331}}}}$$

Divide 1 by \frac{4346}{4331} by multiplying 1 by the reciprocal of \frac{4346}{4331}.

$$3+\frac{1}{7+\frac{1}{15+1\times \frac{4331}{4346}}}$$

Multiply 1 and \frac{4331}{4346} to get \frac{4331}{4346}.

$$3+\frac{1}{7+\frac{1}{15+\frac{4331}{4346}}}$$

Convert 15 to fraction \frac{65190}{4346}.

$$3+\frac{1}{7+\frac{1}{\frac{65190}{4346}+\frac{4331}{4346}}}$$

Since \frac{65190}{4346} and \frac{4331}{4346} have the same denominator, add them by adding their numerators.

$$3+\frac{1}{7+\frac{1}{\frac{65190+4331}{4346}}}$$

Add 65190 and 4331 to get 69521.

$$3+\frac{1}{7+\frac{1}{\frac{69521}{4346}}}$$

Divide 1 by \frac{69521}{4346} by multiplying 1 by the reciprocal of \frac{69521}{4346}.

$$3+\frac{1}{7+1\times \frac{4346}{69521}}$$

Multiply 1 and \frac{4346}{69521} to get \frac{4346}{69521}.

$$3+\frac{1}{7+\frac{4346}{69521}}$$

Convert 7 to fraction \frac{486647}{69521}.

$$3+\frac{1}{\frac{486647}{69521}+\frac{4346}{69521}}$$

Since \frac{486647}{69521} and \frac{4346}{69521} have the same denominator, add them by adding their numerators.

$$3+\frac{1}{\frac{486647+4346}{69521}}$$

Add 486647 and 4346 to get 490993.

$$3+\frac{1}{\frac{490993}{69521}}$$

Divide 1 by \frac{490993}{69521} by multiplying 1 by the reciprocal of \frac{490993}{69521}.

$$3+1\times \frac{69521}{490993}$$

Multiply 1 and \frac{69521}{490993} to get \frac{69521}{490993}.

$$3+\frac{69521}{490993}$$

Convert 3 to fraction \frac{1472979}{490993}.

$$\frac{1472979}{490993}+\frac{69521}{490993}$$

Since \frac{1472979}{490993} and \frac{69521}{490993} have the same denominator, add them by adding their numerators.

$$\frac{1472979+69521}{490993}$$

Add 1472979 and 69521 to get 1542500.

$$\frac{1542500}{490993}$$

Factor

$$\frac{2 ^ {2} \cdot 5 ^ {4} \cdot 617}{490993} = 3\frac{69521}{490993} = 3.141592649997047$$