Problem $2$: $(10$ points$)$ Based on Problem $1,$ assume that $a_n$ is a sequence

that is recursively defined and represents $n^{th}$ chapter or episode. Also we can

define $P(n)$ as true when $a_n$ makes since based on the background information.

If you don't want to use your answer in Problem $1,$ you may use a different

example $($including either example listed above$).$ Answer the following questions

below:

$($a$)$ What are the initial values of the system? These are the elements in the

sequence that do not have dependencies. List out $a_1,a_2,a_3,$dots,$a_t.$ For

example, in the Marvel Movies, initial values include Thor, Iron Man, and

Doctor Strange.

$($b$)$ Choose some $a_k$ and list out all the immediate dependencies for this spe$-$

cific chapter or episode. For example, let $a_k$ be Civil War. This depends

on Age of Ultron and Ant Man.

$($c$)$ List all the recursive dependencies for $a_k.$

$($d$)$ What would make $P(k)$ false if one of the dendencies were not met? List

a few examples. For example, if we watched Civil War before The Winter

Soldier, it might be hard to understand what they are fighting about.

Similarily, if Ant Man was not watched, the giant man in the Avenger

battle would be confusing.

$($e$)$ What would you consider $a_{k+1}$ and how does it rely on $a_k$ and other

dependencies $a_1,a_2,$dots,$a_k?$

$($f$)$ Come up with your own sequential ordering for reading or watching the

series. Remember that the ordering must satistify the dependencies for

$P(n)$ to be true. In other words, list all $a_1,a_2,$dots,$a_n.$ What order do you

recommend reading or watching the series in and why?