Question $2.$ The following inductive proof establishes that Fn $2$n for n $0$:

$($Base cases.$)$ For n $=0$ and n $=1,$ observe that F$0=01=20$ and F$1=12=21,$

establishing our base cases that Fn $2$n for n $=0,1.$

$($Inductive step.$)$ Fix n $2,$ and assume Fi $2$i for all i $=0,...,$ n $1.$ Note that

Fn $=$ Fn$2+$ Fn$1$ by definition, and Fn$22$n$2$ and Fn$12$n$1$ by assumption. Then

Fn $2$n$2+2$n$12$n$1+2$n$1=2$n$,$ establishing the desired conclusion for n $2.$

Provide a similar proof that Fn $2$n$/2$ for n $6.$ Consider what base cases you need!