Question

Question #47095

Answer 1

Sorry for the inconvenience
(this answer is based on a mathematical trick and not explanation )

Explanation:

`Answer=2`

The trick for this:

If you see a question lie adding the roots of roots of a same number to infinite times,Take out the number in it (in this case is `2`)

And make the number in the form

`n(n+1)=2`

If we solve for it, we get `n=1,n+1=2`

So,`n+1` will be the solution

Some examples

`sqrt(12+sqrt(12+sqrt(12+sqrt(12))))...=4`

`sqrt(72+sqrt(72+sqrt(72+sqrt(72))))...=9`

Answer 2

`sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + … ∞)))) = 2`

Explanation:

`sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + … ))))`

This is an amazing concept which is quite simple and easy
So look carefully
Lets say
`sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + … ∞))) )= x`

Now square both sides

`2 +color(red)(" sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + … ∞))) )) = x^2`

Now if you notice the part under our radical is the same as under our first equation;

`color(red)sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + … ∞)))) = x`

Lets plug the x where it belong in the second equation

`2 + x = x^2`

Right now i think we are getting somewhere

`x^2 - x - 2= 0`

Factor this

`(x - 2)(x + 1 ) = 0`

`x = 2 or -1`

Lets remind ourselves

`sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + … ∞)))) = x`

Clearly this is positive

So `x != -1`

Hence

`sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + … ∞)))) = x = 2`