Need help with 4.1 and 4.2. Need mathematical proofs

ll T-Mobile Wi-Fi the 4:28 PM order of reach xfundamentals of algorith...E Problem 4.1. How much time does the following "algorithm require as a function for t ? 1 to n do for J-1to?do for k 1 to n de Express your answer in notation in the simplest possible form. You may consider that each individual instruction (including loop control) is elementary material 140 Analysis of Algorithms Chapter 4 Problem 42 How much time does the following algorithm" require as a function of m? for 1 to n do for j-1 to i do for k-j to n do Express your answer in 8 notation in the simplest possible form. You may consider that each individual instruction (including loop control) is elementary Problem 4.a. Consider the loop for i 1 to m do P which is part of a larger algorithm that works on an instance of size . Let bethe time needed for each execution of P, which we assume independent of i for the sake of this problem (but t could depend on n). Prove that this loop takes a time in ?( mt ) provided t is bounded below by some constant and provided there exists a threshold no such that m z 1 whenever n no. (Recall that we saw in Section 4.2.2 that the desired conclusion would not hold without those restrictions) Problem 4.4. Prove by mathematical induction that the values of i and j at the end of the k-th iteration of Fibiter in Section 4.22 are fi-1 and fi. respectively where fa denotes the n-th Fibonacci number Problem 4.5. Consider the following algorithm to compute binomial coefficients function C(n,k) if k-0 ork-n then return 1 else return C(n-1,k-1+C(n-1.k) Analyse the time taken by this algorithm under the (unreasonable) assumption that the addition CIN-1,k-1)+C(n-1,k) can be carried out in constant time once both Cin-1.k-1) and C(n-1,k) have been obtained recursively. Let tn) denote the worst time that a call on C(n, k) may take for all possible values of k 0 s k s n. Express t(n) in the simplest possible form in ? notation