Question

Find X

$x = \sqrt{6 \sqrt{ 6 \sqrt{6 \sqrt{6 } } }.....}$

Answer 1

√(6+√(6+√(6+…..+ )))
Let X = √(6+√(6+…..+ ))
Then, X = √(6+ X)
Square root on both side,
X² = 6 + X
X² – X – 6 = 0
X² – 3 X + 2 X – 6 = 0
X (X – 3) + 2 (X – 3) = 0
(X + 2) (X – 3) = 0
X = 3, – 2
But – 2 < 0  therefore, X = 3.
Then, the value of √(6+√(6+√(6+…..+ ))) is 3.