Φ = sqrt(1+sqrt(1+sqrt(1+sqrt(1+…
Φ^2 = 1+sqrt(1+sqrt(1+sqrt(1+…
Φ^2 - 1 = sqrt(1+sqrt(1+sqrt(1+…
And using the very first line,
Φ^2 - 1 = Φ
Φ^2 - Φ - 1 = 0
Using the quadratic formula,
Φ = (1 +/- sqrt(1^2 - 4(1)(-1)))/(2)
Φ = (1 +/- sqrt(1+ 4))/(2)
Φ = (1 +/- sqrt(5))/(2)