\( \sqrt {x+\sqrt {x+\sqrt {x+\sqrt {x+\sqrt {x+\sqrt {x.....}}}}}} \)

This fascinating expression defines a recursive relationship! If we let \( y = \sqrt {x+\sqrt {x+\sqrt {x+\sqrt {x+\sqrt {x+\sqrt {x.....}}}}}} \), we can notice that the inner expression is actually the same as \( y \) itself. Therefore, we can rewrite it as: \[ y = \sqrt{x+y} \] By squaring both sides, we get: \[ y^2 = x + y \] Rearranging this gives us the quadratic equation: \[ y^2 - y - x = 0 \] Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -1, c = -x \), we can solve for \( y \) as: \[ y = \frac{1 \pm \sqrt{1 + 4x}}{2} \] Since \( y \) must be non-negative, we choose the positive root: \[ y = \frac{1 + \sqrt{1 + 4x}}{2} \] This is the result of the nested radical! Now, for some real-world applications: this kind of infinite radical is often used in mathematical modeling, such as looking at recursive processes in computer science or predicting growth trends in economics. It showcases how seemingly complex problems can unravel simply through algebraic manipulation!