BeMath-x-x-x-x-4-4-4-4-x-

Question Number 107484 by bemath last updated on 11/Aug/20

∦ B e M a t h ∦

\sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \dots}}}} = \sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \dots}}}}

x = ?

Answered by john santu last updated on 11/Aug/20

⋇ JS ⋇

\Rightarrow x + \sqrt{x + \sqrt{x + \sqrt{x + \dots}}} = 4 \sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \dots}}}}

\Rightarrow x + \sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \dots}}}}} = 4 \sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \sqrt{\dots}}}}}

(1) l e t t = \sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \dots}}}}}}

t^{2} = 4 t \Rightarrow t = 4

(2) x + 4 = 4.4 \Rightarrow x = 12

Answered by Dwaipayan Shikari last updated on 11/Aug/20

\sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x \dots .}}}} = p

x + p = p^{2}

p^{2} - p - x = 0

p = \frac{1 + \sqrt{1 + 4 x}}{2}

\sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4}}}}}} = a

4 a = a^{2}

a = 4

1 + \sqrt{1 + 4 x} = 8

1 + 4 x = 49

x = 12