Answer
See below.
Work Step by Step
1. Let $P(n)$ be the statement to be proved.
2. Test for $n=0,1,2,3$, we have $c_0=2,c_1=2, c_2=6, c_3=3c_0=6$ all are even, thus $P(0), P(1),P(2),P(3)$ are true.
3. Suppose it is true for $n\le p, (p\gt3)$, that is $ c_p, c_{p-1} ...$ are even.
4. For $n=p+1$, let $c_{p-2}=2m$ (m is an integer), we have $c_{p+1}=3c_{p-2}=6m$ which is also even..
5. Thus $P(p+1)$ will also be true and we have proved the statement using mathematical induction.