Answer
See below.
Work Step by Step
1. Let $P(n)$ be the statement to be proved.
2. Test for $n=1,2,3$, we have $b_1=4, b_2=12, b_3=4+12=16$ all are divisible by 4, thus $P(1),P(2),P(3)$ are true.
3. Suppose it is true for $n\le p, (p\gt3)$, that is $ b_p, b_{p-1} ...$ are divisible by 4.
4. For $n=p+1$, let $b_{p}=4k, b_{p-1}=4m$ (k,m are integers), we have $b_{p+1}=b_{p-1}+b_{p}=4k+4m=4(k+m)$ which is also divisible by 4..
5. Thus $P(p+1)$ will also be true and we have proved the statement using mathematical induction.