Formal proof,$$f_{n+2}=f_{n+1}+f_{n}$$For $n=1,$
$f_{3}=f_{2}+f_{1} = 1 + 1 = 2$ is an even number
and $f_{3},$ here $3\%3=0$
Hence, the statement is true for $n=1$
Now, let this statement is true for $n=k$
So, $f_{k+2}=f_{k+1}+f_{k},$ where $f_k$ and $f_{k+1}$ is $\text{ODD}$ and $f_{k+2}$ is $\text{EVEN},$ and also $(k+2)\%3=0$
Now, Prove this statement true for $(k+5)$ which is next to $3^{rd}$ number after $(k+2)$
$f_{k+3}=f_{k+2}+f_{k+1} = \text{ODD}(\text{EVEN} + \text{ODD})$
$f_{k+4}=f_{k+3}+f_{k+2}= \text{ODD}(\text{ODD} + \text{EVEN})$
$f_{k+5}=f_{k+4}+f_{k+3}= \text{EVEN}(\text{ODD} + \text{ODD})$
Hence, it is also true for $(k+5).$
So$,$ By the principle of mathematical induction $f(n), $ is true for all $n\%3=0$