# A Fibonacci Sum

Probability Level 5

A sequence $\{a_i\}_{i=1}^n$ has the property that $S_n=F_n$ where $S_n=\displaystyle\sum^n_{i=1}a_i$, $F_1=1$, $F_2=1$, and $F_n=F_{n-1}+F_{n-2}$. The closed form of $\displaystyle\sum^n_{i=1}a_{2i-1}$ can be represented as $S_{f(n)}+c$ where $f(n)$ is a function of $n$ and $c$ is a constant. Find the last three digits of $f(2013)+c$.

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