7. Prove that if a, = 1, a, = 4, and a, = 6a,-1 – 9an-2 for n 2 2, then a, = 3"-(3+ n) for all n 2 0. Show all your algebraic steps! Basic case(s): Inductive Hypothesis: Inductive step - we need to prove:

= For integer n ≥ 1 let P(n) be the predicate that 9" - 5n CEZ 4c for some For the induction hypothesis, consider k ≥ 1, and suppose that P(k) is true. For the inductive step, we want to show that P(k+1) is true. True or false: The following proof correctly proves P(k+ 1) true, where every step other than the one labelled IH follows by algebra. (I'm asking: is this a valid algebraic proof? Is the algebra correct? Did I use the IH correctly? Did I get the correct final result?) 9k+15k+19.9k - 5.5k True False = 9(4c – 5k) – 5 ⋅ 5% for CE Z by the IH = 4.9c-9.5k - 5.5k = 4.9c-4.5k = 4d for d = Z

This is Discreet math please answer questions with proper formatting for proofs

Show that PNP CE₂. (In fact, also PNP C II and NPNP = 2. Once you've proved the first fact, these two are easier to show, but you don't have to show them for this exercise.) Hint: You need to simulate an oracle machine with an 3 quantifier and a quantifier. Use the 3 quantifier to guess oracle answers and both quantifiers to check the truthfulness of the answer.

Pick one million sets of 12 uniform random numbers between 0 and 1. Sum up the 12 numbers in each set. Make a histogram with these one million sums, picking some reasonable binning. You will find that the mean is (obviously?) 12 times 0.5 = 6. Perhaps more surprising, you will find that the distribution of these sums looks very much Gaussian (a "Bell Curve"). This is an example of the "Central Limit Theorem", which says that the distribution of the sum of many random variables approaches the Gaussian distribution even when the individual variables are not gaussianly distributed. mean Superimpose on the histogram an appropriately normalized Gaussian distribution of 6 and standard deviation o = 1. (Look at the solutions from the week 5 discussion session for some help, if you need it). You will find that this Gaussian works pretty well. Not for credit but for thinking: why o = 1 in this case? (An explanation will come once the solutions are posted).

Formally prove or disprove the following claims, using any method

Prove by induction that there exists a knight's walk of an n-by-n chessboard for any n > 4. (It turns out that knight's tours exist for all even n > 6, but you don't need to prove this fact.)

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).]

Use strong induction to prove the following:

3. Let p(n) = Σª a¡n², where a¿ > 0, be a degree-d polynomial in n, and let k be a constant. Prove the following properties. if k ≥ d, then p(n) = 0(nk); if k ≤d, then p(n) = N(nk); if k = d, then p(n) = 0(nk); if k > d, then p(n) = o(n²); if k < d, then p(n) = w(nk); a. b. C. d. e.