Factorial of a number is defined as:    n! = n(n-1)(n-2)(n-3)...(2)(1)   For example, 4! = 4*3*2*1     The n! can be written in terms of (n-1)! as:   n! = n* (n-1)!   (n-1)! = (n-1)*(n-2)!   and so forth.     Thus, in order to compute n!, we need (n-1)!, to have (n-1)!, we need (n-2)! and so forth. As you may immediately notice, the base case for factorial is 1 because 1! = 1. Write a program that uses a recursive function called factorial that takes an integer n as its argument and returns n! to the main.

Q: Use mathematical induction to prove that the statements are true for every positive integer (a)…

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Q: f(n) such that f(n) ∈ O(n √ n) and f(n) ∈ Ω(n log n)) but f(n) ∈/ Θ(n √ n) and f(n) ∈/ Θ(n log n)).…

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A: For n from 1 to 4, i didn't see any pattern, so I changed it to 16.

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A: According to the Question below the Solution:

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Q: Prove or Disprove: n 7 = O (7 n

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A: Hi there, Please find your solution below, I hope you would find my solution useful and helpful.…

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Q: What is the time complexity of the following algorithm? Select one: a. Θ(n/2) b. Θ(n) c. Θ(n2) d.…

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Q: 10. 1²+2²+...+n² integer n ≥ 1. = n(n+1)(2n +1) 6 for every

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Q: Consider f(n)=  n2 log (n),    g(n)= 0.01 n + 0.5 n2 which of the following is true? Theta = θ…

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Q: Using whatever method you want, find and prove the answers to the following- a) T(n) = 3T(n /3) +…

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Q: Choose the appropriate portion to complete the algorithm written to find this summation x2 x+ x6 x8…

A: The correct option is given in step 2.

Q: Q) Why log2(n) is less time complex as compared to n2? Justify your answer by giving valid examples.

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Q: 5.What is the worst-case efficiency class of the following algorithm?* line no. 1 ALGORITHM Q5…

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Q: Prove that if n is an integer, then ⌊n/2⌋ = n/2 if n is even and (n − 1)/2 if n is odd.

A: Actually, the answer has given below;

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Q: Let p(n) 3+3-5+3 5++3-5-3(5+1-1)/4 a) What is the statement P(1)? b) Complete the inductive step of…

A: a)  P(1) = 18 b) The statement is true for all natural numbers.

Q: 2. The derivation of Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, .. (FIB(N – 1) + FIB(N – 2) If N> 2…

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Q: Given f(n) E O(n), prove that f(n) E 0(n²). Given f(n) E 0(n) and g(n) E O(n²), prove that f(n)g(n)…

A: Prove the given asymptotic

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Q: Prove the following, or give a counter example: (a) f(n) — О(g(n)) and g(n) — - О(h(m)). O(h(n))…

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Q: Compare the order of growth of 1. 2^n and n! 2. log |n| and √n

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Q: Problem 1. True or False? Justify your answer briefly. a. n 3 + 2n 2 ∈ ω(0.1n 3.1 + n) b. n log5 n…

A:

Q: For each of the following expressions below, write the corresponding Big-Oh complexity Expression…

A: 1. O(2n) 2. O(n7)

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A: The explanation is given in next step As per Bartleby rules we are answering the first 3 questions.