Question

(5+(5+(5+......)^1/2)^1/2)^1/2

Answer 1

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Answer 2

question is -----> Evaluate : $\sqrt{5+\sqrt{5+\sqrt{5+......}}}$


$x = \sqrt{5+\sqrt{5+\sqrt{5+......}}} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{x}$
or, $x=\sqrt{5+x}$

squaring both sides,

x² = 5 + x

x² - x - 5 = 0

use quadratic formula to find value of x.

x = $\frac{-(-1)\pm\sqrt{(-1)^2-4(-5)(1)}}{2(1)}$

x= $\frac{1\pm\sqrt{21}}{2}$

but here x ≥ 0 [ because every square root function is positive function. ]

so, x ≠ $\frac{1-\sqrt{21}}{2}$

hence, x = $\frac{1+\sqrt{21}}{2}$