Problem 1:
For the Fibonacci matrix
Fibonacci numbers by matrix multiplication (due to Knuth): m+1 F 1 1 n
2) In class we showed a Matrix algorithm for Fibonacci numbers: 1 112 1 0 n+1 F (Note: No credit for an induction proof that this is true. I'm not asking that.) a) What is the running time for this algorithm? (3 pts.) b) Prove it. (9 pts.)
Question 3 Program Language C++
Problem 3 Fibonacci Numbers 10 points Fibonacci numbers are a sequence of numbers where each number is represented by the sum of the two preceding numbers, starting with 0 and 1: 0, 1, 1, 2, 3, 5, 8, etc Write a program that repeatedly prompts the user for a positive integer and prints out whether or not that integer is a Fibonacci number. The program terminates when-I is entered. Create a method with the following...
discrete math
Problem 7.8 (Explore: Fibonacci Identities). The Fibonacci numbers are a famous integer sequence: Fn) o 0, 1, 1,2,3, 5, 8, 13, 21, 34, 55, 89,... defined recursively by Fo 0, F1, and F F Fn-2 for n2 2. (a) Find the partial sums Fo+Fi +F2, Fo+ Fi +F2Fs, Fo + Fi + F2+Fs +F, FoF1+F2+ Fs+F4F (b) Compare your partial sums above with the terms of the Fibonacci sequence. Do you see any patterns? Make a conjecture for...
Problem 2: (8 pts) The Fibonacci sequence is the series of numbers 0, 1, 1, 2, 3, 5, 8.,.. Formally, it can be expressed as: fib0-0 fibl-1 fibn-fibn-1+fibn-2 Write a multithreaded program that generates the Fibonacci sequence. This program should work as follows: On the command line, the user will enter the number of Fibonacci numbers that the program is to generate. The program will then create a separate thread that will generate the Fibonacci numbers, placing the sequence in...
Problem 7 ii (Explore Fibonacci Partial Sums). Let F. 에 be the Fibonacci sequence. (a) Find the partial sums Fo + Fi +Po, Fo+Fİ +B+F3. Fo +Fi+B+F +ћ. Fo + Fi +B+B+F+E, (b) Compare your partial sums above with the terms of the Fibonacci sequence. Do you see any patterns? Make a conjecture for Fo+ Fi+Fs and Fo+Fo. Decide if your conjecture is true by actually computing the sums. Revise your conjecture if necessary. (c) Make a conjecture for Fo...
(4) A Fibonacci-like sequence Gk is constructed by setting G0 =
0, G1 = 1, and Gk+2 = 1 2 (Gk+1 + Gk). (a) Find a matrix A so that
Gk+2 Gk+1 = A Gk+1 Gk . (b) Diagonalize A and find a formula of Ak
in terms of k. (c) Use this to find a formula for Gk. (d) Find
limk→∞ Gk.
(4) A Fibonacci-like sequence Gk is constructed by setting Go 0, G1 1, and Gk+2(GkG a) Find...
(5) Fibonacci sequences in groups. The Fibonacci numbers Fn are defined recursively by Fo 0, F1 -1, and Fn - Fn-1+Fn-2 forn 2 2. The definition of this sequence only depends on a binary operation. Since every group comes with a binary operation, we can define Fibonacci- type sequences in any group. Let G be a group, and define the sequence {fn in G as follows: Let ao, a1 be elements of G, and define fo-ao, fi-a1, and fn-an-1an-2 forn...
c++ fibonacci code using loops
Here are 8 Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21 Note that the first Fibonacci number is 1, F(1) = 1 The second Fibonacci number is 1, i.e. F(2) = 1 Other Fibonacci numbers in the sequence is the sum of two previous Fibonacci numbers. For example F(3) = F(2) + F(1). In general F(n) = F(n-1) + F(n-2) Write a program to do the following tasks. User entries are shown in...
Need help making this Java program: package assignment1; public class Fibonacci { // Exercise 1: Fibonacci numbers // // fibonacci(n) returns nth Fibonacci number, and is defined by the // recurrence relation F_n = F_n-1 + F_n-2, with seed values F_0=0 and F_1=1. public long getFibonacci(long number) { } }