$1(0\mathrm{.}5$ points$)$ The following is the initial adjacency matrix for the Floyd algorithm. Proof that

$D^{(1)}[5][2]=4$ using a Dynamic Programming approach. Show all your work to support your

proof.

$2.$ Using sequential search:

The probability that the item IS in the array is $3$

$4.$ If the item is in the array, the probability

that the last item in the array matches the search key is $1$

$3,$ and probability that the next to

the last item of the array matches the search key is $1$

$4,$ and the probability is found at position

$($n $2)$ is $1$

$5.$ The probabilities matching any of the remaining items are all equal.

A$.$What is the probability $($as a function of n$)$ of matching one of the $1$st through

$($n $3)$nd items? Show all your work to get credit

B$.$Assuming that comparison of an array item where the search key is the basic

operation, what is the average case complexity function for sequential search under these

conditions. Show all your work to get credit

C$.$What is the big O average case time complexity under these conditions.

$3.$ Compute the complexity and Big O time complexity $($worst case$)$ of the following

pseudocode assuming the print statement is the basic operation. Show all your work step by

step to get credit including summations.

repeat n times $\{$

for $($i$=$n; i$>1$; i$=$i$/2)\{$

for $($j$=1$; $$j