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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