Let $a_1=F_{k-1}-c$ for some integer $c$ $a_2=F_{k+1}-a_1=F_k+c$ $a_3=F_{k+2}-a_2=F_{k+1}-c$ And by induction we can prove $a_n = F_{n+k-2} + (-1)^nc$

Lemma: $\frac{F_{n+k-2} - a_n}{(-1)^{n-1}}$ is constant $\forall$ $n \geq 1$. Proof: It's like to prove that $\frac{F_{n+k-2} - a_n}{(-1)^{n-1}}$ $= \frac{F_{n+k-1} - a_{n+1}}{(-1)^n}$ $\Longleftrightarrow$ $F_{n+k-2} - a_n = $$\frac{F_{n+k-1} - a_{n+1}}{(-1)}$ $\Longleftrightarrow$ $F_{n+k-2} + F_{n+k+1} $$= a_{n+1} + a_n$ $\Longleftrightarrow F_{n+k} = F_{n+k}$ Let $c = \frac{F_{n+k-2} - a_n}{(-1)^{n-1}}$.