Prove: Vn € Z¹, 3(12)” + 8 is divisible by 11.

Prave that if 3nt4 is then odd.

7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).

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2. Prove that 8"-3" is evenly divisible by 5 for all natural numbers n.

Prove or disprove that if f(n) = (g(n)), then 4f(n) = (49(n)).

Prove that 14n is not O(12n)

suppose a,b, and c are odd integers. Prove that a+b+c is odd

Show that StartFraction d Over dx EndFraction ​(ln kx​)equalsStartFraction d Over dx EndFraction ln x​, given that xgreater than 0 and kgreater than 0 is a real number.

Solve T(n) = 64T(n/8) - n2 log(n) using master theorem