$(1$ point$)$

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 inner loop. This total will depend on $i.$

In each iteration of the inner loop,

operations are performed.

The inner loop runs

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

With the information above, we can count the number of operations the overall algorithm computes.

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

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

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 this loop is

$=$

$($start adding from the smallest value of $i)$

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

$.$