Fun with Fibonacci

Algebra Level 3

Let $$F_{n}$$ be an infinite sequence, in which $$F_{1} = F_{2} = 1$$, and satisfies the recurrence relation $$F_{n} = F_{n-1} + F_{n - 2}$$, for positive integers $$n \geq 3$$.

Let $$S$$ be an infinte series in which $$S$$ is defined as

$$\Large\displaystyle\sum_{n = 1}^{\infty} \frac{F_{n}}{2^{n}}$$

Calculate the value of $$S$$

This problem appeared in the AITMO 2013

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