= Solution
Use <mathematical induction>, starting at $n=1$. Since $F_0=0$ and $F_1=1$, the sum is $-1=(-1)^1F_0-1$.
Suppose the formula holds at $n\geq1$. Adding the next <Fibonacci number> with its sign and using the <Fibonacci recurrence> gives
$$
\begin{aligned}
\sum_{k=0}^{n+1}(-1)^kF_k
&=(-1)^nF_{n-1}-1+(-1)^{n+1}F_{n+1}\\
&=(-1)^{n+1}(F_{n+1}-F_{n-1})-1\\
&=(-1)^{n+1}F_n-1.
\end{aligned}
$$
This is the required successor case. Thus
$$
\boxed{\sum_{k=0}^{n}(-1)^kF_k=(-1)^nF_{n-1}-1\quad(n\geq1)}.
$$
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