points$)$ Continue finding $k(\%)$ until the sequence eventually gets to $1.$ How many times did it take?

$(0\mathrm{.}5$ points$)$ Try it again, but this time start with $m=8.$ How long did it take for the sequence to get to $1?$

You could set up a loop so that you wouldn't have to do each of these by hand. Try

$>m$:$=3$:

$>k(m)$:

$>$ for $j$ from $1to200$ while $k(\%)>1dok(\%)$: end $do$:

Define $m$ to be $3.$

Plug $m$ into $k(x).$

Create a loop that runs while $k(x)>1.$

$>pr{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(m,j+1)$

Print out $m$ and the number of steps, $j+1.$

The Collatz Conjecture predicts that each sequence will eventually get to $1$ after some finite amount of steps.

a$)(2$ points$)$ Can you prove this is true $($exhaustively$)$ for all starting integers $m60?$

Hint: Set up a loop so that you don't have to individually check each value of $m.$

b$)(1$ point$)$ Suppose you change the $3x+1$ to $x+1$ in the definition of $k(x).$ What happens? Can you prove it$?$