Solve for x: x = 1 + sqrt(1+sqrt(1+sqrt(1+ ... )))?

Answer 1

Let's call this sum 1+s, where s is the square root part.

x = 1+s

But we know that s^2 is the same as 1+s (the stuff inside the first square root is identical to the original sum).

s^2 = 1 + s
s^2 - s - 1 = 0

Using the quadratic:

s = (1 +/- sqrt(1 + 4))/2
s = (1 + sqrt(5))/2

Since x=1+s, and ignoring the negative solution:

x = 1 + (1+sqrt(5))/2
x = 3/2 + sqrt(5)/2

Answer 2

4

Answer 3

xxx+2sqrt(1-1)=0-sex<4U