Question 4: The function T(n) is recursively defined as follows: 1 if n = 1, A + A T(n – 1) if n 2 2. Prove that T(n) = 0(n log n).
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Q: Write a recursive mathematical definition for computing 2n for a positive integer n.
A: Given: Write a recursive mathematical definition for computing 2n for a positive integer n.
Q: Examples: Find the tight bound of the following recursive functions, T(n). T(n)=9T(n/3)+n T(n)=…
A: the solution is an given below :
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A: I'm providing the answer to the above question. I hope this will be helpful for you.
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A: Basis step: For n = 1 F1F2 = (1)(1) = 1 = 12 = F12 The proposition is true for n = 1.
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Q: Write a recursive mathematical definition for computing for a positive integer n.
A: Given: Write a recursive mathematical definition for computing for a positive integer n.
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A: Given: To write the recursive formula.
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- r Examples: Find the tight bound of the following recursive functions, T(n). T(n)=9T(n/3) +n T(n)= T(2n/3) +1 ● T(n)=3T(n/4)+nlogn ● T(n) = 2T (n/2)+nlogn T(n)=2T(n/2) + O(n) • T(n)=8T(n/2) +0(n²) ● T(n) = 7T(n/2) + O(n²)8- Determine if each of the following recursive definition is a valid recursive definition of a function f from a set of non-negative integers. If f is well defined, find a formula for f(n) where n is non- negative and prove that your formula is valid. a. f(0) = 2,f(1) = 3, f(n) = f(n-1)-1 for n ≥ 2 b. f(0) = 1,f(1) = 2, f(n) = 2f (n-2) for n = 21. Compute the recursive function f(n) = 3f(n–3)– l; ƒ(0) = 1; for n=12;. Show %3D %3D %3D clearly how the function is evaluated and the return of the function for every recursive step as seen in class. Show all your work to get credit
- Your main task is to write a recursive function sierpinski() that plots a Sierpinski triangle of order n to standard drawing. Think recursively: sierpinski() should draw one filled equilateral triangle (pointed downwards) and then call itself recursively three times (with an appropriate stopping condition). It should draw 1 filled triangle for n = 1; 4 filled triangles for n = 2; and 13 filled triangles for n = 3; and so forth. Sierpinski.java When writing your program, exercise modular design by organizing it into four functions, as specified in the following API: public class Sierpinski { // Height of an equilateral triangle with the specified side length. public static double height(double length) // Draws a filled equilateral triangle with the specified side length // whose bottom vertex is (x, y). public static void filledTriangle(double x, double y, double length) // Draws a Sierpinski triangle of order n, such that the largest filled //…Let F be the function such that F(n) is the sum of the first n positive integers. Give a recursive definition of F(n).Question: Let t(x) be the number of primes that areIn programming, a recursive function calls itself. The classical example is factorial(n), which can be defined recursively as n*factorial(n-1). Nonethessless, it is important to take note that a recursive function should have a terminating condition (or base case), in the case of factorial, factorial(0)=1. Hence, the full definition is: factorial(n) = 1, for n = 0 factorial(n) = n * factorial(n-1), for all n > 1 For example, suppose n = 5: // Recursive call factorial(5) = 5 * factorial(4) factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) factorial(2) = 2 * factorial(1) factorial(1) = 1 * factorial(0) factorial(0) = 1 // Base case // Unwinding factorial(1) = 1 * 1 = 1 factorial(2) = 2 * 1 = 2 factorial(3) = 3 * 2 = 6 factorial(4) = 4 * 6 = 24 factorial(5) = 5 * 24 = 120 (DONE) Exercise (Factorial) (Recursive): Write a recursive method called factorial() to compute the factorial of the given integer. public static int factorial(int n) The recursive algorithm is:…The Polish mathematician Wacław Sierpiński described the pattern in 1915, but it has appeared in Italian art since the 13th century. Though the Sierpinski triangle looks complex, it can be generated with a short recursive function. Your main task is to write a recursive function sierpinski() that plots a Sierpinski triangle of order n to standard drawing. Think recursively: sierpinski() should draw one filled equilateral triangle (pointed downwards) and then call itself recursively three times (with an appropriate stopping condition). It should draw 1 filled triangle for n = 1; 4 filled triangles for n = 2; and 13 filled triangles for n = 3; and so forth. API specification. When writing your program, exercise modular design by organizing it into four functions, as specified in the following API: public class Sierpinski { // Height of an equilateral triangle whose sides are of the specified length. public static double height(double length) // Draws a filled equilateral…8. Give a recursive definition of the sequence {an}, n =1, 2, 3, ... ifa) an = 4n − 2. b) an = 1 + (−1)n.c) an = n(n + 1). d) an = n2.Consider the following recursive function: if b = 0, if 6 > a > 0, a f(b, a) f (b, 2.(a mod b)) otherwise. f(a, b) = Estimate the number of recursive applications required to compute f(a, b).if n = 1 S(n) S(n//2) S(n – 1) if n is even; // is Python integer division if n is odd and n >1 Does Master Theorem apply to the recursive function S? Answer by writing either "yes" or "no" There is no need for extra explanations in this part.(a) Give a recursive definition of F(n) where F(n) =1+2+3+....+n. (b) Find the value of a4 if a1 = 1, a2 = 2, and an =an−1 + an−2 +· · ·+a1SEE MORE QUESTIONS