MATLAB

Problem $6(20$ Points$)$

The Collatz Conjecture is an interesting one. Consider the following operation on an arbitrary

positive integer from $n$ to $f(n)$ :

$F(n)=\{\begin{array}{ccc}3n+1,& nis\text{o}\text{d}\text{d}& \\ \frac{n}{2},& nis\text{e}\text{v}\text{e}\text{n}& \end{array}$

That is$,$ if the number is even, divide it by two; if the number is odd, triple it and add one. For

example, by applying this operation successively for a starting point $n=3,$ we get:

$3|10|5|16|8|4|2|1|4|2|1$cdots

It was conjectured that, starting with any positive integer, this operation eventually ends up with

the periodic sequence $4|2|1.$ There is no proof yet!

a$)$ Write a user$-$defined function "Collatz" which has $n$ as input, and the following outputs:

A vector with all integers in the sequence.

The number of steps such that the number $1$ first appears in the sequence.

The maximum integer that the sequence reached for a given value $n.$

b$)$ For the set of integers in the interval $2,1000,$ write a code which calls the user$-$defined

function and produces the following $($and show the outputs with appropriate details and labels$)$:

A plot for the sequence corresponding to $n=731.$

A plot of $n$ vs the sequence length.

A plot of $n$ vs the maximum integer in its corresponding sequence.

A plot of $n$ vs the average of consecutive differences of the sequence integers.