Question

What is the value of √2√2✓2✓2......

Answer 1

Heya user,

Consider the equation, y = √[2√[2√[2√[2√[2√[2√[2 ... ∞
Then,
-----> $\frac {y^2}{ 2} = \sqrt{2} \sqrt{2} \sqrt{2} \sqrt{2} \sqrt{2} ...$
-----> y² / 2 = y
-----> y² = 2y
===>  y ( y - 2 ) = 0

But since, y > √2, y cannot be zero --> Hence, y = 2 is the soln.

And therefore, √[ 2 √ [ 2 √ [ 2 ... ∞ = 2

Answer 2

Let √2√2√2...... be x [eq. 1]

[Squaring both sides]

(√2√2√2....)² = x²

2(√2√2√2...) = x²

2(x) = x² [From eq. 1]

x=2

So, √2√2√2... = 2