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Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer
This is a homework question so I'm looking to just be nudged in the right direction, I'm not asking for my work to be done for me.
The Fibonacci numbers are defined as follows: $f_0 = 0$, $f_1 = 1$, and $f_{n+2} = f_n + f_{n+1}$ whenever $n \geq 0$. Prove that when $n$ is a positive integer:
\begin{equation*} f_0 - f_1 + f_2 + \ldots - f_{2n-1} + f_{2n} = f_{2n-1} - 1 \end{equation*}
So as I understand it, this is an induction problem. I've done the basis step using $n = 1$:
\begin{align*} - f_{2(1)-1} + f_{2(1)} &= f_{2(1)-1} - 1\newline - f_1 + f_2 &= f_1 - 1\newline - 1 + 1 &= 1 - 1\newline 0 &= 0 \end{align*}
I've concluded that the inductive hypothesis is that $- f_{2n-1} + f_{2n} = f_{2n-1} - 1$ is true for some $n \geq 1$. From what I can gather, the inductive step is:
\begin{equation*} f_0 - f_1 + f_2 + \ldots - f_{2n-1} + f_{2n} - f_{2n+1} = f_{2n} - 1 \end{equation*}
However, what I find when I try to prove it using that equation is that it is incorrect. For example, when I take $n = 1$
\begin{align*} - f_{2(1)-1} + f_{2(1)} + f_{2(1)+1} &\neq f_{2(1)} - 1\newline - f_1 + f_2 - f_3 &\neq f_2 - 1\newline - 1 + 1 - 2 &\neq 1 - 1\newline -2 &\neq 0 \end{align*}
I suppose that my inductive step is wrong but I'm not sure where I went wrong. Maybe I went wrong elsewhere. Any hints?