problem: What is the value of $x$ for which $$\sqrt{x + \sqrt{x + \sqrt{x + \ldots}}} = 5?$$
solution: Squaring both sides of the given equation yields $$x + \sqrt{x + \sqrt{x + \ldots}} = 25,$$ so $$\sqrt{x + \sqrt{x + \ldots}} = 25-x.$$ We already know the value of the left-hand side to be equal to $5$. Thus, $5 = 25-x$, and $x = \boxed{20}.$