The value of $3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}$ is equal to

Let $$x = 3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}$$<br><br>So, $x = 3 + {1 \over {4 + {1 \over x}}} = 3 + {1 \over {{{4x + 1} \over x}}}$<br><br>$\Rightarrow (x - 3) = {x \over {(4x + 1)}}$<br><br>$\Rightarrow (4x + 1)(x - 3) = x$<br><br>$\Rightarrow 4{x^2} - 12x + x - 3 = x$<br><br>$\Rightarrow 4{x^2} - 12x - 3 = 0$<br><br>$$x = {{12 \pm \sqrt {{{(12)}^2} + 12 \times 4} } \over {2 \times 4}} = {{12 \pm \sqrt {12(16)} } \over 8}$$<br><br>$= {{12 \pm 4 \times 2\sqrt 3 } \over 8} = {{3 \pm 2\sqrt 3 } \over 2}$<br><br>$x = {3 \over 2} \pm \sqrt 3 = 1.5 \pm \sqrt 3$.<br><br>But only positive value is accepted<br><br>So, x = $1.5 + \sqrt 3$