The value of 4+frac{1}{5+frac{1}{4+frac{1}{5+ | vedclass

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Question

(A) Let $y = 4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\ldots}}}$.Observe that the expression repeats after the first two terms: $y = 4+\frac{1}{5+\frac{1}{y

Vedclass · Accepted Answer

$2+\frac{2}{5} \sqrt{30}$

Vedclass · Answer

(A) Let $y = 4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\ldots}}}$.Observe that the expression repeats after the first two terms: $y = 4+\frac{1}{5+\frac{1}{y}}$.Simplify the equation: $y - 4 = \frac{1}{\frac{5y+1}{y}} = \frac{y}{5y+1}$.Cross-multiply: $(y-4)(5y+1) = y$.$5y^2 + y - 20y - 4 = y$.$5y^2 - 20y - 4 = 0$.Using the quadratic formula $y = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$:$y = \frac{20 \pm \sqrt{(-20)^2 - 4(5)(-4)}}{2(5)} = \frac{20 \pm \sqrt{400 + 80}}{10} = \frac{20 \pm \sqrt{480}}{10}$.Since $y > 0$,we take the positive root: $y = \frac{20 + \sqrt{480}}{10} = 2 + \frac{\sqrt{16 	imes 30}}{10} = 2 + \frac{4\sqrt{30}}{10} = 2 + \frac{2\sqrt{30}}{5} = 2 + \frac{2}{5}\sqrt{30}$.

The value of $4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\ldots \ldots \infty}}}}$ is

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