Calculate the value of $\frac{1}{2}+\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+....+\frac{F_n}{2^n}+....$ The Fibonacci sequence starts with 1, 1, 2, 3, 5, 8, 13, ... .(Start from the 3rd term,
each term is the sum of the two previous terms). Let $F_n$ be the $n$th term of this sequence. $S$ is defined as $S=\frac{1}{2}+\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+....+\frac{F_n}{2^n}+....$
Calculate the value of $S$
I have no idea how to solve this, hints aswell as solutions would be appreciated 
Taken from the 2013 AITMO

$$F_n=\frac{a^n-b^n}{\sqrt{5}},~ a+b=1, ~ab=-1,~ a,b=\frac{1\pm\sqrt{5}}{2}.$$
The required sim $$ s=\sum_{n=1}^{\infty} \frac{F_n}{2^n}= \frac{1}{\sqrt{5}} \sum_{n=1}^{\infty} \left ( \frac{a^n}{2^n}-\frac{b^n}{2^n} \right) =\frac{1}{\sqrt{5}}\left(\frac{a}{2-a}-\frac{b}{2-b}\right)= \frac{1}{\sqrt{5}}~\frac{2(a-b)}{4-2(a+b)+ab}=\frac{2 \sqrt{5}}{\sqrt{5}}=2.$$
