Problem $1.(4+8=12$ points$)$ What does this code do$?$ You encounter the following mysterious piece of code.

Algorithm $1$: Mystery Function$$Function F$($n$))$: If n $=0$ :

Return $(2,1)$ Else

b$1$

Fori from$1$ton b $2$b

$($u$,$v$)$F$($n$1)$ Return $($u $+$ b$,$ v $$ b$)($a$)$ What are the results of F$(1),$ F$(2),$ F$(3),$ and F$(4)?$ Solution:

$($b$)$ What does the code do in general, when given input integer n $>=0?$ Prove your assertion by induction on n$.$

Solution: