The value of 2 + frac{1}{2 + frac{1}{2 + dots | vedclass

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(B) Let $x = 2 + \frac{1}{2 + \frac{1}{2 + \dots \infty}}$.Since the expression is infinite,we can substitute the repeating part with $x$:$x = 2 + \fr

Vedclass · Accepted Answer

$1 + \sqrt{2}$

Vedclass · Answer

(B) Let $x = 2 + \frac{1}{2 + \frac{1}{2 + \dots \infty}}$.Since the expression is infinite,we can substitute the repeating part with $x$:$x = 2 + \frac{1}{x}$.Multiplying both sides by $x$,we get $x^2 = 2x + 1$,which simplifies to the quadratic equation $x^2 - 2x - 1 = 0$.Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$,where $a = 1, b = -2, c = -1$:$x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-1)}}{2(1)} = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2}$.Since the expression $2 + \frac{1}{2 + \dots}$ must be positive and greater than $2$,we reject $1 - \sqrt{2}$ (which is negative).Therefore,the value is $1 + \sqrt{2}$.

The value of $2 + \frac{1}{2 + \frac{1}{2 + \dots \infty}}$ is

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