$(1$ point$)$

$($Challenge problem!$)$

Assume $n$ is a positive integer. Our goal is to find a power function $g(n)=n^p$ such that the algorithm below is $O(n^p).$

Start by counting the number of operations performed in the innermost loop $($the $k-$loop$).$ This total will depend on $j.$

In each iteration of the innermost loop,

operations are performed.

The inner loop runs

times, so the total number of operations performed in this loop is

Now we want to count the number of operations the middle loop $($the $j-$loop$)$ performs in a complete run of it$.$

In the $j^{th}$ iteration of the middle loop,

operations are performed. $($This total depends on the value of $j.)$

The middle loop runs a total of

times.

To get the total number of operations, we need to add up the number of operations performed for each value of $j.$ That is$,$ the total number of operations performed in the middle loop is

$($Start adding from the smallest value of $j.)$

Finally, we compute the total number of operations this algorithm performs

In the $i^{th}$ iteration of the outer loop,

operations are performed. $($This total depends on the value of $i.)$

The outer loop runs a total of

times.

To get the total number of operations, we need to add up the number of operations performed for each value of $i.$ That is$,$ the total number of operations performed in the algorithm is

$+$

$+$cdots$+=$

$($Start adding from the smallest value of $i.)$

Therefore, by the order of a polynomial theorem, this algorithm is $O($

$.$