\(2015\) 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 \(\alpha + \beta + \gamma\)

Please show your solutions!

Image credit: Flickr Jean
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