1. Prove that n? +1 > 2n, where n is a positive integer with 1

Q: Let a be an integer such that a = 4 mod 12. Use the definition of mod to prove that a² = 4 mod 12-

A: Solution-:                           a≡4 mod 12a2=(4)2 mod 12     =16 mod  12     =4   mod   12

Q: Given T1(n)=O(f(n)) and T2(n)=C(g(n)). Find T1(n).T2(n)

A: Given :- The time complexity of two function f(n) and g(n) is mention in the above given question…

Q: 2- Prove thatthe following statements are true (7) or false (F). (let log n = log₂n). You must…

A: The complete answer is below:

Q: 1.) Please prove the following using definitions. a.) 2n4 − 5n2 ∈ Θ(n4) b.) n log n − n ∈ Ω(n log…

A: Proved the given functions of the asymptotic notations

Q: Prove the following statement by contraposition. For every integer x, if 5x 2 – 2x + 1 is even, then…

A: Proof for every integer x, if 5x2 – 2x + 1 is even, then x is odd.

Q: : a. [(a mod n) - (b mod n)] mod n = (a - b) mod n

A:

Q: Let a and b be integers. Then a divides b if and only if: O ab-k for som integer k O you can't…

A: Answer is in next step.

Q: There are n students who studied at a late-night study for final exam. The time has come to order…

A: When ordering pizzas for a group of students studying for their final exams, it's essential to…

Q: Let n EN be a natural number with a divisor d E N. If the sum of all of the positive divisors of n…

A: #note: Since programming language is not specified, we are providing answer in python #function to…

Q: for ( i = 1 ; i < n ; i * 2 ){         for ( j = 0 ; j < min( i , k ) ; j++){             sum++…

A: Time complexity = O(n^2)

Q: An evil king has n bottles of wine, and a spy has just poisoned one of them. Unfortunately, they do…

A: This is a conventional information theory dilemma requiring a binary explore algorithm to solve. The…

Q: Solve the following recurrences using iteration methods and Master's Theorem (if possible) a. T(n) =…

A: Hello studentGreetingsRecurrence relations are equations that describe the time complexity of…

Q: Prove: For all integers n, if n² is odd, then n is odd. Use a proof by contraposition, as in Lemma…

A: Contraposition Method : contraposition is a kind of immediate inference during which a proposition…

Q: 15- Prove that for every positive integer n: 1·2· 3 + 2·3 - 4+ . +n(n + 1)(n + 2) = n(n + 1)(n +…

A: principle of mathematical induction.    Given p(n) checking for n=1,    LHS:1.2.3=6…

Q: Master theorem: if n<d T(n)= { |aT(n/b)+ f (n) if n2 d 1. if f(n) is O(n08, «-e 2. if f(n) is O(no%,…

A: Master Theorem:

Q: Let n be an integer. If 3n+ 4 is odd, then n is odd.

A: even*even=even even*odd=even odd*even=even odd*odd=odd   odd+even=odd even+odd=odd even+even=even…

Q: uppose that f(n) = 0(g(n)) and f(n) = 0(h(n)), then it is ( always / sometimes / never) the case…

A: Defined the given statement always, sometimes or never

Q: Explain, with an example why the following definition, would not be suitable or useful: f(n) is Ω(…

A: This is about time complexity.

Q: Find the first three examples of an odd number x>0 and an even number y>0 such that x − y = 7.

A: In step 2, I have provided  ANSWER  with BRIEF  EXPLANATION---------------

Q: 7. Prove or disprove: f(n) + g(n) = 0 (min(f(n), g(n)))

A: Let ,f(n) = n2g(n) = n