QUESTION 26 What is the time complexity function of the following algorithm? 1. Input string ₁,..., n, n and s 2. For i = 1 to n do 2.1. If as then 2.1.1. Return "String contains s." 3. Return "String does not contain s." a. f(n)=n b. f(n)=n² f(n) = n²_n 2 d. f(n) = 2n

1. Draw the flowchart and trace table for the exponential function shown below. The Algorithm should request from the user the value of "x" and the number of iterations (approximation-stopping point). Both should be positive integers. For the factorial, you can use the built-in function, or in other words, you can represent the factorial (n!) as a Predefined Process 1=1+x+x²+x²³...

Solve the following Integer Programming problem using the graphical or branch and bound algorithm, where n=5, s=13.

5. Find the generating function to determine the number of ways to pick k objects from n objects when repetitions are allowed and the ith object appears i or 2 times for 1≤ i ≤n.

6. Discover the recursive formula for Hn. Find Hn for n=1,2,3...6. Based on these values you will be able tosee the recursive pattern. proving the formula is optional for extra credit. Consider all subsets of N={1, 2, 3,..., n). How many of these subsets, SC N, satisfy the property that (i) Vx (x ES → x +1 ¢ S) ? Denote the number of subsets satisfying this property (i) by Hn. (a) Find the first 6 values of Hn . I have calculated n=1 and n=2, so you need to do n=3, 4, 5, 6 (b) Find and prove the recursive formula for the sequence Hn . Example: Let n=1. Then N={1} and the subsets satisfying property (i) are {1} and O so H1=2. Let n=2. Then N={1,2} and the subsets satisfying property (i) are {1}, {2}, and Ø, i.e. H3=3. Note: {1,2} is also subset of {1,2} but since for x=1, x+1=2 and both 1 and 2 are contained in {1,2} therefore the set {1,2} fails to satisfy property (i). 7. Prove by induction that the number of subsets for a set with n elements is 2^n.

Please include justifications for each step.