OK, I am completely stupid when it comes to induction. Any help? I shouldn't have taken this class.
Use Mathematicla induction for part 1.
let $\displaystyle S_n $be a statement that $\displaystyle a_n$ is odd
Assume $\displaystyle S_{k-2}$ and $\displaystyle S_{k-1}$ are true and then prove $\displaystyle S_k $is true.
$\displaystyle a_k = 2a_{k-2} +3a_{k-1} $
Now $\displaystyle 2a_{k-2}$ is even and $\displaystyle 3a_{k-1}$ is odd since odd multiplied by odd is odd.
Sum of an even number and odd number is always odd.
so $\displaystyle a_k $is odd.
Hence $\displaystyle S_k$ is proved true.
base case $\displaystyle a_1$ and $\displaystyle a_2 $are already give odd.