Loop Invariants Consider the procedure SUM-ARRAY below. It computes the sum of n numbers n the array A[1..n]. State a loop invariant for this procedure, and use its nitialization, maintenance, and termination properties to show that the SUMARRAY procedure returns the sum of the numbers in A[1..n]. SUM-ARRAY(A,n)1sum=0fori=1tonsum=sum+A[i]returnsum Inversions and Insertion-Sort Consider the insertion-sort algorithm given below (from CLRS): INSERTION-SORT (A,n) 12345678fori=2tonkey=A[i]//InsertA[i]intothesortedsubarrayA[1:i1].j=i1whilej>0andA[j]>keyA[j+1]=A[j]j=j1A[j+1]=key Let A[1..n] be an array of n distinct numbers. If IA[j], then the pair (i,j) is called an inversion of A. What is the relationship between the running time of insertion sort and the number of inversions in the input array? Justify your answer. Asymptotic notations. Consider the following functions: f(n)=n! g(n)=nlognh(n)=2n State the set of (6) relationships among them in terms of O() and (). Show your reasoning using the definitions of O() and ()