How do you write script to calculate the value of x in the following nested square root for different numbers of square roots

lets say x = sqrt(2+sqrt(2+sqrt2+sqrt2+........

how would i be able to write a script that could calculate it? I eventually it would converge to a certain number.

All help is appreciate because i dont even know the first place to start Thanks!

Answer 1

[Image Analyst]

Ah, it's the old, famous trick recursive equation question. There are lots of these. But what you need to realize is that since x is that, then x is also

x = sqrt(2 + x)

Squaring both sides

x^2 = 2 + x

or

x^2 - x - 2 = 0

Solving for the roots:

>> roots([1, -1, -2])
ans =
     2
    -1

Other versions can be solved with the same concept - just realize the equation is in there as a "subequation" somewhere, so just replace it with x and solve for x. Easy/trivial once you know the "trick".