Given that: \(\frac{1}{{1 + \frac{1}{{1 + \frac{1}{{1 + \frac{1}{x}}}}}}} = \frac{5}{8}\) then what is the value of x?
2
Simplify the nested fraction from inside out:
\[1+\tfrac{1}{x}=\tfrac{x+1}{x};\quad 1+\tfrac{x}{x+1}=\tfrac{2x+1}{x+1};\quad 1+\tfrac{x+1}{2x+1}=\tfrac{3x+2}{2x+1}\]
So the LHS equals \(\dfrac{2x+1}{3x+2}=\dfrac{5}{8}\). Cross-multiplying:
\[8(2x+1)=5(3x+2)\Rightarrow 16x+8=15x+10\Rightarrow x=\mathbf{2}\]
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