Solve $\dfrac{1}{1+\frac{1}{1+\ddots}}$ I'm currently a high school junior enrolling in AP Calculus, I found this website that's full of "math geeks" and I hope you can give me some clues on how to solve this problem.  I'm pretty desperate for this since I'm only about $0.4%$ to an A- and I can't really afford a B now...
The problem is to simplify:  $$\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}}$$
What I did, was using basic "limits" taught in class and I figured out that the denominator would just keep going like this and approaches $1$, so this whole thing equals $1$, but I think it's not that easy...

This is the Golden ratio (also known as $\varphi$) expressed using countinued fraction. This number is solution of the $x^2-x-1=0$ quadratic equotation.
This quadratic equotation you can wrote as $x=1+\frac{1}{x}$ and this form is used to construct cotinued faction used in your question.
See wikipedia for "Golden Ratio" or read book writed by Mario Livio: Golden ratio for example. Here you found more answers.
