<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] You can see that the expression is found within itself. The trick is to manipulate that fact...
$$N+1 = 2 + \frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+...}}}}}$$
$$N+1 = 2+\frac{1}{N+1}$$
Can you solve the quadratic and continue from here?
P.S. You will get two values for $N$ by solving the quadratic. Only one of them is correct...
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