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 $a_1=1, a_2=3, a_3=1+2(3)=7$, thus $P(1),P(2),P(3)$ are true.
3. Suppose it is true for $n\le p, (p\gt3)$.
4. For $n=p+1$, let $a_{p}=2k+1, a_{p-1}=2m+1$ (k,m are integers), we have $a_{p+1}=a_{p-1}+2a_{p}=2m+1+4k+2=2(2k+m+1)+1$ which is also odd.
5. Thus $P(p+1)$ will also be true and we have proved the statement using mathematical induction.