$\frac{1}{1+\frac{1}{1+\frac{1}{9}}}$
Answer (1)
Simplify the innermost fraction: 1 + 9 1 = 9 10 .
Invert the result: 9 10 1 = 10 9 .
Add 1: 1 + 10 9 = 10 19 .
Invert again: 10 19 1 = 19 10 .
The final answer is 19 10 .
Explanation
Understanding the Problem We are asked to evaluate the continued fraction 1 + 1 + 9 1 1 1 . To do this, we will simplify the fraction step by step, starting from the innermost fraction.
Simplifying the Innermost Fraction First, we simplify the innermost fraction: 1 + 9 1 . To add these, we need a common denominator, which is 9. So we have: 1 + 9 1 = 9 9 + 9 1 = 9 9 + 1 = 9 10
Inverting the Fraction Next, we take the reciprocal of the result from the previous step: 1 + 9 1 1 = 9 10 1 = 10 9
Adding 1 Now, we add 1 to the result from the previous step: 1 + 1 + 9 1 1 = 1 + 10 9 = 10 10 + 10 9 = 10 10 + 9 = 10 19
Inverting Again Finally, we take the reciprocal of the result from the previous step: 1 + 1 + 9 1 1 1 = 10 19 1 = 19 10
Final Answer Therefore, the value of the continued fraction is 19 10 .
Examples
Continued fractions can be used to approximate real numbers. For example, they are used in computer science to represent floating-point numbers and in number theory to study Diophantine approximations. They also appear in the study of dynamical systems and chaos theory, where they can be used to analyze the behavior of certain types of maps.
