# $$2015$$ and Fibonacci

Calculus Level 5

Let $$a_1$$, $$a_2$$, $$a_3$$, $$\dots$$, $$a_{2014}$$, and $$a_{2015}$$ be real numbers and also let $f\left( x \right) =\begin{cases}x + a_1 \qquad \qquad \qquad \qquad \qquad x \le 1 \\2x + a_2 \, \, \, \quad \qquad \qquad \qquad \qquad 1 < x \le 2 \\ 3x + 2a_3 \quad \qquad \qquad \qquad \qquad 2 < x \le 3 \\\vdots \\ 2014x + F_{2014}a_{2014} \, \, \, \, \, \, \, \quad \qquad \, \, \, \, 2013 < x \le 2014 \\2015x + F_{2015}a_{2015} \, \, \, \, \, \, \, \, \, \, \, \quad \qquad 2014 < x \le 2015 \\ 2016x + F_{2016}a_{1} \, \, \, \, \, \, \quad \quad \quad\qquad x > 2015 \end{cases}$ where $$F_n$$ is the $$n^{th}$$ Fibonacci Number.

If $$f\left( x \right)$$ is continuous everywhere, then the maximum value of $$a_1 + a_2 + a_3 + \dots + a_{2014} + a_{2015}$$ can be expressed as $-\frac { \alpha }{ F_\beta - F_\gamma - 1 }$ where $$\alpha$$, $$\beta$$, and $$\gamma$$ are positive integers.

Find the value of $$\alpha + \beta + \gamma$$