Prove the following statement using mathematical induction: For all integers n ≥ 0, 3 | ((n^3) + 8n)

Let P(n) be the statement: “10n – 1 is divisible by 11, where n is a non-zero even natural number” Use mathematical induction to prove P(n). You must show all the steps of the proof using mathematical induction. Show your work step by step.

TRUE OR FALSE N Queens' problem can be effectively solved using dynamic programming.

Devise a recursive algorithm for finding n! mod m where n and m are positive integers.

a)Write a recursive definition for the set of odd positive integers. b)Use master theorem to find the solution to the recurrence relation f(n) = 4f(n/2) + 2? ! ,when n = 2" , where k is a positive integer and f(1) = 1.

Question 5 Please answer them correctly by choosing one of the following answer

Consider the following recursive algorithm Algorithm Q(n)// Input: a positive integer nif n == 1 return 1else return Q(n-1)+2*n-1 1) Set up a recurrence relation for this function’s value and solve it to determine what this algorithm computes. 2) Setup a recurrence relation for the number of multiplication made by this algorithm and solve it to find the total number of multiplications that are executed.

Use induction to show that 4n < 2nfor all n ≥5

Order the following functions by asymptotic growth rate (number 1 is the best algorithm, and number 3 is the worst). 4nlog n+2n 2log n n³ + 2

The n-th harmonic is the sum of the reciprocals of the first n natural numbers given by: 17 1.1 1 1 H₁ =Σ =1+=+=+ - 2 3 4 k=1 k +. + n (i) Write a recursive algorithm of this function returning, H, with nε N. (ii) Give a recurrence relation for the number of divisions the recursive algorithm calculates for an input n & N. iii) Solve the recurrence relation and give the O(n) of the algorithm.