This is one of the more challenging problems promised in your syllabus. It looks pretty bewildering but it can be solved by a straightforward application of one of the main principles we discuss in this class every day. Suppose $$x = \sqrt{1+ \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{ 1 + \ldots}}}}}$$ where the square roots go on forever. You may assume (and it is true) that this expression actually defines a positive number $x$. What is it? Enter it as a decimal or radical expression: