Question : If $2=x+\frac{1}{1+\frac{1}{5+\frac{1}{2}}}$, then the value of $x$ is equal to:
Option 1: $\frac{14}{13}$
Option 2: 1
Option 3: $\frac{13}{15}$
Option 4: $\frac{15}{13}$
Correct Answer: $\frac{15}{13}$
Solution :
Given: The expression is $2=x+\frac{1}{1+\frac{1}{5+\frac{1}{2}}}$.
⇒ $2=x+\frac{1}{1+\frac{1}{\frac{10+1}{2}}}$
⇒ $2=x+\frac{1}{1+\frac{1}{\frac{11}{2}}}$
⇒ $2=x+\frac{1}{1+\frac{2}{11}}$
⇒ $2=x+\frac{1}{\frac{11+2}{11}}$
⇒ $2=x+\frac{1}{\frac{13}{11}}$
⇒ $2=x+\frac{11}{13}$
⇒ $x=2–\frac{11}{13}$
⇒ $x=\frac{26–11}{13}=\frac{15}{13}$
Hence, the correct answer is $\frac{15}{13}$.
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