Evaluate $1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{...}}}}$ when you see $15$ fraction lines Evaluate $1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}}$ when you see $15$ fraction lines.
I have solved this problem but using a quater calculating I come from down to up 15 times and found the answer $\frac{987}{610}$. And the time of exam was an hour It should be an easier way to solve it.That takes about 5 minutes.
update1:The last fraction is:$1+\frac{1}{1}$
 A: After the first two or three steps you might notice that numerator and denominator are consecutive Fibonacci numbers. Let $\frac{a_n}{b_n}$ be the value with $n$ horizontal lines. Then
$$\frac{a_{n+1}}{b_{n+1}}=1+\frac1{a_n/b_n}=1+\frac{b_n}{a_n}=\frac{a_n+b_n}{a_n}\;:$$
the new numerator is the sum of the old numerator and denominator, and the new denominator is the old numerator. Thus, 
$$\frac{a_n}{b_n}=\frac{F_{n+1}}{F_n}\;,$$
and you can easily calculate the first $16$ Fibonacci numbers by hand using the recurrence $F_n=F_{n-1}+F_{n-2}$. Alternatively, you can use the fact that $F_n$ is the integer closest to
$$\frac{\varphi^n}{\sqrt5}\;,$$
where $\varphi=\frac12\left(1+\sqrt5\right)$.
A: Let $f(x) = 1 + \frac 1x$.  We denote $f^k(x) = \overbrace{f(\cdots(f}^{k}(x))\cdots)$.
It would seem, then, that you are calculating $f^{15}(1)$.  Note that
$$
f(a/b) = 1 + \frac ab = \frac{a + b}{b}.  
$$
Thus, we can calculate $f^{15}(1)$ efficiently by the process
$$
\frac 11 \mapsto \frac{2}{1} \mapsto \frac 32 \mapsto \frac 53 \mapsto \cdots \mapsto \frac{987}{610} \mapsto \frac{1597}{987}
$$ 
this process is made even easier if you happen to know the Fibonacci numbers.
