Problem $1.$ 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

blarr$1$

For i from $1$ to $n$

$\frac{L}{b}ar(blarr2b)$

$(u,v)larrF(n-1)$

Return $(u+b,v*b)$

$($a$)$ What are the results of $F(1),F(2),F(3),$ and $F(4)?$

Solution:

Type your solution here.

$($b$)$ What does the code do in general, when given input integer $n0?$ Prove your assertion

by induction on $n.$

Solution: Type your solution here.