Consider the sequence F$1,$ F$2,$ F$3,...,$ where

F$1=1,$ F$2=1,$ F$3=2,$ F$4=3,$ F$5=5$ and F$6=8.$

The terms of this sequence are called Fibonacci numbers.

$($a$)$ Define the sequence of Fibonacci numbers by means of a recurrence relation.

$($b$)$ Prove that $2|$ Fn if and only if $3|$ n$.$