I'm trying to work on the problem below, though I've hit a wall on how to proceed to prove the inductive step.
Suppose that $c_0,c_1,c_2\ldots$ is a sequence defined as follows: $$c_0=2,\, c_1=2,\, c_2=6,\, c_k=3c_{k-3}\,(k\geq3)$$
Prove that $c_n$ is even for each integer $n\geq0$.
Here is what I have so far:
Show that $P(0)$ and $P(1)$ are true.
- $c_0=2$ and $2\geq0$ and $2\mid2$, so this is even.
- $c_1=2$ and $2\geq0$ and $2\mid2$, so this is even.
Show that for every integer $k\geq1$, if $P(i)$ is true for each integer $i$ with $0\leq i\leq k$, then $P(k+1)$ is true.
- Let $k$ be any integer with $k\geq1$, and suppose $c_i$ is even for each integer $i$ with $0\leq i\leq k$ [inductive hypothesis].
- I must show that $c_{k+1}$ is even for each integer $k\geq0$.
- Now $c_{k+1}=3c_{k-2}$...
...and this is where I do not understand how to proceed. Any tips on how I can finish solving this problem are greatly appreciated.