Proof

\[\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{...}}}}=x\]

\[x+\sqrt{x+\sqrt{x+\sqrt{...}}}=x^2\]

\[\sqrt{x+\sqrt{x+\sqrt{...}}}=x^2-x\]

\[x=x^2-x\]

\[0=x^2-2x\]

\[0=(x)(x-2)\]

equation <- function(x, iter) {
  if (iter == 0) {
    return(sqrt(x))
  } else {
    return(sqrt(x+equation(x, iter-1)))
  }
}

equation(0,100)

[1] 0

equation(2,100)

[1] 2

Q.E.D.

Original Question

Experimental Solution

# equation(0.1,100)
# equation(0.5,100)
equation(1, 100)

[1] 1.618034

There is our good friend \[\phi\] again!

Keep increasing X until we get the answer …

equation(1.1, 100)

[1] 1.661895

# ...
# equation(1.8, 100)
# equation(1.99, 100)
equation(2, 100)

[1] 2

Make sure it isn’t true for any other values higher than 2 ?

equation(2.1, 100)

[1] 2.032971

equation(3, 100)

[1] 2.302776