Following, we can consider the number f(n) of perfect matchings in a 3 × 2n grid graph, which is the same as the number of ways to tile a 3 × 2n rectangle with 1 × 2 dominoes. a) Prove that f(0) = 1,f(1) = 3, and that for positive n, f(n) =3f(n−1)+2f(n−2)+ ...+2f(0) b)  Prove(probablyusingtheformulain(a))thatforn>1,f(n)=4f(n−1)−f(n−2). c)Prove by induction (probably using the formula in (b)) that for all naturals n, f(n) = ((1 + 1/√3)(2 + √3)n) + (1 − 1/√3)(2 − √3)n))/2 d)Using any of these formulas, find f(n) for all n ≤ 5

Q: Let g(n) = 13 + g(n) = 0(n4) giving the constants. + 23 + + n3. Show that %3D

A: According to principle methemetical induction 13 + 23 + 33 + … + n3  = (n(n+1)2)2 On Solving this we…

Q: 5. Let g(n) = log10 (n). Prove that g(n) = (lgn). Please show step by step solution. Show the work.

A: Given that, g(n)=log10(n) If we consider this as recurrence relation in the form aT(n/b)+nklogpn…

Q: Prove or disprove that for any x ∈ N, x(x+1)/2 ∈ N (where N = {0, 1, 2, 3, ….}

A: Hello student

Q: how that f(n)= n2+2n+1 is O (n2).  and also show that f(n)= n2+2n+1 is not O(n). Compute c an

A: Proving Big-Oh: Example 1 Show that f (n) = n2 + 2n + 1 is O(n2). Choose k = 1. Assuming n > 1,…

Q: You are standing in front of an infinitely long straight fence; that is, the fence extends…

A: Solution is given below :

Q: True/False? Let f(n)=O(g(n)) and g(n)=O(T(n)). Is this true that f(n)=O(T(n))? Prove your claim and…

A: Asymptotic notation is a mathematical framework used to describe the behavior of functions as their…

Q: An electrician has wired n lights, all initially on, so that: 1) light 1 can always be turned…

A: IMPLEMENTATION: The algorithm works from left to right implementing individual bits. First bit can…

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: You are given a N*N maze with a rat placed at maze[0][0]. Find whether any path exist that rat can…

A: The Rat in a Maze problem is a classic problem in the field of computer science and algorithm…

Q: Expand the following recurrence to help you find a closed-form solution, and then use induction to…

A: This problem can be solved by a recursive algorithm. It is an equation that specifies the values of…

Q: Prove that the following shows LHS = RHS : (+ 2 1 ) = 3 Given: 1=λf.λy.f y 2=λf.λy.f(f y)…

A: Hello student Greetings Hope you are doing great. Thank you!!!

Q: (b) Prove: max(f(n), g(n)) E O(f(n)g(n)), i.e., s(n) E O(f(n)g(n)). Assume for all natural n, f(n) >…

A: Given that, f(n)>=1 and g(n)>=1 s(n)= max(f(n), g(n))

Q: Heuristics Prove or disprove: If h1(n), ..., hk(n) are admissible, so is h(n) = h1(n) + ... + hk(n)

A: It is generally understood that the pointwise maximum of any set of admissible heuristics h1,...,hk…

Q: 4. Consider f(n) = 3n2 + 4n – 3, mathematically show that f(n) is O(n?), 2(n²), and O(n2).

A:

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: with n=6 and A=(3,5,4,1,3,2). Draw the corresponding walkthrough as shown in P.158

A: In this question, we have to sort elements using bubble sort. And we have to draw walkthrough of…

Q: Suppose, T(n) = 12+22+32+…..+n2. Show that, T(n) is in Θ(n3 ).

A: Given  T(n) = 12+22+32+…..+n2

Q: You are given a N*N maze with a rat placed at maze[0][0]. Find whether any path exist that rat can…

A: I have done the code using Java language, Make sure you are on latest version of Java.

Q: For n > 1, in how many out of the n! permutations 7 = (7(1), 7(2),., "(n)) of the numb {1, 2, ...,…

A: Hey there, I am writing the required solution of the questin mentioned above. Please do find the…

Q: 2) Prove divisible by 3 for any integer nzo. that n(n²+s) is divisible by 3 for Let n 1 1 (1² +5) =…

A: Please see the solutions.Explanation: Step 3: Step 4:

Q: Prove that the sum of the first n odd positive integers is n2. In other words, show that 1 + 3 + 5 +…

A: To prove that the sum of the first n odd positive integers is n^2, we will use mathematical…

Q: GIVEN: n red points and n blue points in the plane in general position (i.e., no 3 points are on the…

A: For such problems, implementing this using the pigeonhole principle completes the work   Therefore,…

Q: Determine φ (m), for m=12,15, 26, according to the definition: Check for each positive integer n…

A:

Q: 2. Strictly Increasing Given a function, f(x), which is a strictly increasing function, define an…

A: Algorithm that finds the smallest positive integer, n, at which the function, f(n), becomes…

Q: If f(n) = O(g(n)), then g(n) = Ω(f(n))

A: Given : - The functions f(n) and g(n) be functions with domain {1, 2, 3, . . .} Asymptotic analysis…

Q: et G: (0,1)" (0,1)"+l be defined as follows: (X1. ....x,) = (X, x, OX2, X OX2 O X3,X x2 ... x, X).…

A:

Q: Show that T(n) = 0(n²) where T(n) = 1+ 2 + 3+. .. +n

A: We have to show that T(n) = O(n2)  Where T(n) = 1+2+3+.......+n

Q: Let A = {(N1, N2) | N1 and N2 are NFAS and L(N1)N L(N2) = Ø}. Show that A is %3D decidable.

A: Given that, A={ <N1, N2> | N1 and N2 are NFAs and L(N1)∩L(N2)=∅} That means N1 and N2 are…

Q: . Find ƒ (1), ƒ (2), ƒ (3), ƒ (4), and ƒ (5) if f (n) is defined recursively by f(0) = 3 and for n =…

A: Here in this question we have given two functions and we have asked to find the value of f(1) ,…

Q: Prove by induction that fib(0)+fib(1)+…+fib(n) = fib(n+2)-1, fib(0)=0 fib(1)=1

A: Fibonacci numbersThere is a close connection between induction and recursive definitions: induction…

Q: Expand the following recurrence to help you find a closed-form solution, and then use induction to…

A: Here, the master method is appropriate. The recursion tree is quite useful. The iteration method…