Given the following recursive algorithm:sumCube(n)Input: positive integer nOutput: 13 +23 + ... + n³ = 11³if n=1 return 1else return n³ + sumCube (n-1)=Prove that sumCube(n) returns 13 + 2³ + ... + n³induction. For this proof, answer the following questions:for any positive integer n using Complete the basis step of the proof.What is the inductive hypothesis?What do you need to show in the inductive step of the proof?Complete the inductive step of the proof.

To prove:Recursive algorithm sumCube(n) returns 13+23+33+.....+n3=∑i−1ni3 1. Basis step:when n = 1,…

Answered: Given the following recursive…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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*Find a non-recursive formula for f (n) : f (0) = 7, f (n) = 4f (n − 1)/9 for n ≥ 1 *Define the sequence: an = 1+ (n+3)2 for n = 1,2,3, ... recursively.
Prove that the following recursive algorithmfor computing factorials is correct. function factorial(n) comment Return n!. 1. if n ≤ 1 then return(1) 2. else return(n · factorial(n − 1))
A recursive algorithm has been developed as shown below: Input: List A = (a₁,..., an) such that N > 1 Output: value x foo(A) if length(A) == 1 return a₁ A' = (a₁,..., aN-1) q = foo(A') if (q> an) else return q return an State what problem this algorithm solves and write down the two recurrence equations that characterise this recursive algorithm.
The following recursive procedure computes the n-th Fibonacci number. Fib(n) 1 ifn<= 1 2 return n 3 else return Fib(n - 1) + Fib(n - 2) Use memoization technique to develop an efficient version of Fib(n). Give your solution in pseudocode, similar to what is given above.
ALGORITHM Y (n) // Input: n is a positive integer if n = 1 return 1 else return Y (n − 1) + n ∗ n 1. What does this algorithm compute? 2. Can the algorithm be optimised to give the same output?
Consider the recursive algorithm below for computing the sum of the first n cubes: S(n) = 13 + 23 + … + n3 ALGORITHM S(n) //n is a positive integer if (n==1) return 1 else return S(n – 1) + (n*n*n); Solve the recurrence relation in your answer for the previous question. The runtime efficiency of the algorithm is? A. O(n) B. O(n2) C. O(n3) D. None of the above
Solve the following recurrences using iteration methods and Master's Theorem (if possible) a. T(n) = 2T (n/3) +3 b. T(n) = 3T (n/6) + n
The n-th harmonic is the sum of the reciprocals of the first n natural numbers given by: 17 1.1 1 1 H₁ =Σ =1+=+=+ - 2 3 4 k=1 k +. + n (i) Write a recursive algorithm of this function returning, H, with nε N. (ii) Give a recurrence relation for the number of divisions the recursive algorithm calculates for an input n & N. iii) Solve the recurrence relation and give the O(n) of the algorithm.
Solve by using recursive solutions! no looping statements thank you computePI(int n) – One remarkably simple formula for approximating the value of pi is the so-called Madhava–Leibniz series: pi/4 = 1 − 1/3+1/5 −1/7 +1/9 − ⋯ +/− 1/2n-1. Note that the result of the series is a quarter of pi. The accuracy of the approximation is dependent on the value of n. This method takes an int parameter n and calculates the value of pi, rather than pi/4. Pay attention to the sign (positive or negative) of each term. public double computePI(int n) { // TODO: implement this method return -1; //replace this statement with your own return; }
Write the Fibonacci Function program with: Recursive and Iterative method respectively using the following condition: Fib (1) is 1 Fib (2) is 1 Fib (N) is Fib (N-2) + Fib (N-1), for N > 2 Show the Hand Simulations of activation records for both the programs and display the output
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.
O Make the program that finds the standard deviation (o) for (1- n) x values by using iteration and recursive. o xm=(1/n) E x, O = Vv a (0) = standard deviation V- (1/(n-1 ) Σ ( xm)2 O V = variance value k xm = mean value

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