Question #47095

2 Answers
Mar 3, 2016

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

Mar 3, 2016

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