HMMT 2014 T10

Fix a positive real number $c>1$ and positive integer $n$. Initially, a blackboard contains the numbers $1, c, \ldots, c^{n-1}$. Every minute, Bob chooses two numbers $a, b$ on the board and replaces them with $c a+c^{2} b$. Prove that after $n-1$ minutes, the blackboard contains a single number no less than

$$\left(\frac{c^{n / L}-1}{c^{1 / L}-1}\right)^{L}$$

where $\phi=\frac{1+\sqrt{5}}{2}$ and $L=1+\log _{\phi}(c)$.