The value of 3 + (1)/(4 + (1)/(3 + (1)/(4 + (1)/(3 + ∞)))) is equal...

Let $x = 3 + \dfrac{1}{4 + \dfrac{1}{3 + \dfrac{1}{4 + \dfrac{1}{3 + \ldots \infty}}}}$

So, $x = 3 + \dfrac{1}{4 + \dfrac{1}{x}} = 3 + \dfrac{1}{\dfrac{4x + 1}{x}}$

$⇒ \left(x - 3\right) = \dfrac{x}{( 4x + 1 )}$

$⇒ ( 4x + 1 ) ( x - 3 ) = x$

$⇒ 4{x}^{2} - 12x + x - 3 = x$

$⇒ 4{x}^{2} - 12x - 3 = 0$

$x = \dfrac{12 \pm \sqrt{( 12{)}^{2} + 12 \times 4}}{2 \times 4} = \dfrac{12 \pm \sqrt{12 ( 16 )}}{8}$

$= \dfrac{12 \pm 4 \times 2\sqrt{3}}{8} = \dfrac{3 \pm 2\sqrt{3}}{2}$

$x = \dfrac{3}{2} \pm \sqrt{3} = 1 . 5 \pm \sqrt{3} .$

But only positive value is accepted

So, $x = 1 . 5 + \sqrt{3}$