I had a question regarding the following sum. Let $\phi$ be the golden ratio and $N$ be an even integer.
\begin{array}{lcl} \sum_{n=1}^N (-\phi)^n & = & -\phi + (-\phi)^2 + (-\phi)^3 + \ldots + (-\phi)^N \\ & = & (\phi^2 - \phi) + (\phi^4 - \phi^3) + (\phi^6 - \phi^5) + \ldots + (\phi^N - \phi^{N-1}) \\ &=& \kern 1.5pc 1 \kern 1.16pc + \kern 1.5pc \phi^2 \kern 1.05pc + \kern 1.5pc \phi^4 \kern 1.01pc + \ldots + \kern 1.5pc \phi^{N-2} \end{array}
with $\frac{N}{2}$ terms in the last equation's rhs. I rewrite that as (according to Wolfram)
\begin{array}{lcl} \sum_{n=0}^{N/2 - 1} \phi^{2n} &=& \frac{\phi^N - 1}{\phi^2 - 1} \\ &=& \frac{\phi^N - 1}{\phi} \\ &=& \phi^{N-1} - \phi + 1 \end{array}
The above steps follow from the properties of the golden ratio. But clearly this can't be. Would someone please pinpoint the error in my line of thought?
