<s>[INST] Find $N = 1 + \frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+...}}}}}$ Please show how to solve this step by step, because I don't even have an idea to begin with.
$$N = 1 + \frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+...}}}}}$$
 [/INST] Hint: $$N = 1 + \cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{\ddots}}}}}$$
$$N -1= \cfrac{1}{2+\color{blue}{\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{\ddots}}}}}}$$
Notice that the blue part is also $N-1$.
$$N-1 = \frac{1}{2+\color{blue}{N-1}}$$
$$N-1 = \frac{1}{N+1}$$
It should be simple enough from here. Also, note that by inspection, $N > 1$, so it should be easy to discard an extraneous solution.
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