4+frac{1}{4+frac{1}{4+frac{1}{4+ldots infty}} | vedclass

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Question

(B) Let $x = 4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\ldots \infty}}}$.Since the expression is infinite,we can write $x = 4 + \frac{1}{x}$.Multiplying by $

Vedclass · Accepted Answer

$2+\sqrt{5}$

Vedclass · Answer

(B) Let $x = 4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\ldots \infty}}}$.Since the expression is infinite,we can write $x = 4 + \frac{1}{x}$.Multiplying by $x$,we get $x^2 = 4x + 1$,which simplifies to $x^2 - 4x - 1 = 0$.Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$,we get $x = \frac{4 \pm \sqrt{16 - 4(1)(-1)}}{2} = \frac{4 \pm \sqrt{20}}{2} = \frac{4 \pm 2\sqrt{5}}{2} = 2 \pm \sqrt{5}$.Since the expression $4+\frac{1}{4+\ldots}$ must be positive,we discard $2-\sqrt{5} \approx -0.236$.Thus,$x = 2+\sqrt{5}$.

$4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\ldots \infty}}} = $

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