11. Let n is a positive integer. Prove that n is even if and only if 3n²+1 is odd. (Hint: you must prove bi-implication, it means p q = (p→q) ^ (q→ p).)

2) Prove divisible by 3 for any integer nzo. that n(n²+s) is divisible by 3 for Let n 1 1 (1² +5) = 1(1+5)=(6/ K(K²+5) K+ 1 ((k + 1 ) ² + 5

Prove that the sum of the first n odd positive integers is n2. In other words, show that 1 + 3 + 5 + .... + (2n + 1) = (n + 1)2 for all n ∈ N.

4. Practice with the iteration method. We have already had a recurrence relation ofan algorithm, which is T(n) = 4T(n/2) + n log n. We know T(1) ≤ c.(a) Solve this recurrence relation, i.e., express it as T(n) = O(f(n)), by using the iteration method.Answer:(b)  Prove, by using mathematical induction, that the iteration rule you have observed in 4(a) is correct and you have solved the recurrence relation correctly. [Hint: You can write out the general form of T(n) at the iteration step t, and prove that this form is correct for any iteration step t by using mathematical induction.Then by finding out the eventual number of t and substituting it into your generalform of T(n), you get the O(·) notation of T(n).]

Let f(n) y g(n) two positive asymptotic functions. Prove or disprove the following conjectures: a) f(n) = O(g(n)) implies 2f(n) = O(29(n)). b) f(n) = O(g(n)) implies g(n) = N(f(n)). c) g(n) = O((g(n))²).

Find the value of x for the following sets of congruence using the Chinese remainder theorem.a. x ≡ 2 mod 7, and x ≡ 3 mod 9b. x ≡ 4 mod 5, and x ≡ 10 mod 11c. x ≡ 7 mod 13, and x ≡ 11 mod 12

7. Prove that if 7 divides n with remainder 1, 7 divides n² + 2n +7 with remainder 3.

Give the solution for T(n) in each of the following recurrences. Assume that T(n) is constant for small n. Provide brief justification for your answers.

4. Show that 6n + 3n log(n)= O(n logn). Find the appropriate values of C and no

Let T(n) be defined by the first-order linear recurrence T(n) = 2T(n-1) +8 Suppose it is given that T(2) = c. Compute T(0) by iterating backwards and express your answer in terms of c. T(0) =