Java: An Introduction to Problem Solving and Programming (8th Edition)
8th Edition
ISBN: 9780134462035
Author: Walter Savitch
Publisher: PEARSON
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Textbook Question
Chapter 11, Problem 3PP
The Fibonacci sequence occurs frequently in nature as the growth rate for certain idealized animal populations. The sequence begins with 0 and 1, and each successive Fibonacci number is the sum of the two previous Fibonacci numbers. Hence, the first ten Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34. The third number in the series is 0 + 1, which is 1; the fourth number is 1 + 1, which is 2; the fifth number is 1 + 2, which is 3; and so on.
Besides describing population growth, the sequence can be used to define the form of a spiral. In addition, the ratios of successive Fibonacci numbers in the sequence approach a constant, approximately 1.618, called the “golden mean”. Humans find this ratio so aesthetically pleasing that it is often used to select the length and width rations of rooms and postcards.
Use a recursive formula to define a static method to compute the nth Fibonacci number, given n as an argument. Your method should not use a loop to compute all the Fibonacci numbers up to the desired one, but should be a simple recursive method. Place this static recursive method in a
Fibonacci #1 = 0
Fibonacci #2 = 1
Fibonacci #3 = 1; 1/1 = 1
Fibonacci #4 = 2; 2/1 = 2
Fibonacci #5 = 3; 3/2 = 1.5
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The Fibonacci sequence is listed below: The first and second numbers both start at 1. After that, each number in the series is the sum of the two preceding numbers. Here is an example:
1, 1, 2, 3, 5, 8, 13, 21, ...
If F(n) is the nth value in the sequence, then this definition can be expressed as
F(1) = 1
F(2) = 1
F(3) = 2
F(4) = 3
F(5) = 5
F(6) = 8
F(7) = 13
F(8) = 21
F(n) = F(n - 1) + F(n - 2) for n > 2
Example:
Given n with a value of 4F(4) = F(4-1) + F(4-2)F(4) = F(3) + F(2)F(4) = 2 + 1F(4) = 3
The value of F at position n is defined using the value of F at two smaller positions. Using the definition of the Fibonacci sequence, determine the value of F(10) by using the formula and the sequence. Show the terms in the Fibonacci sequence and show your work for the formula.
A number is called a perfect number if it is equal to the sum of all of its divisors, not including the number itself. For instance, 6 is a perfect number because the divisors of 6 are 1, 2, 3, 6 and 6 = 1 + 2 + 3. As another example, 28 is a perfect number because its divisors are 1, 2, 4, 7, 14, 28 and 28 = 1 + 2 + 4 + 7+ 14. However, 15 is not a perfect number because its divisors are 1, 3, 5, 15 and 15 6= 1 + 3 + 5. Write an algorithm that asks the user to enter a positive integer n and prints all the perfect numbers that are less than n.
A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors. The first perfect number is 6, because 1, 2 and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: (1 + 2 + 3 + 6) / 2 = 6.
Write a program to display the of sum all perfect numbers found in the first 1000 integers.
Chapter 11 Solutions
Java: An Introduction to Problem Solving and Programming (8th Edition)
Ch. 11.1 - What output will be produced by the following...Ch. 11.1 - What is the output produced by the following code?Ch. 11.1 - Write a recursive definition for the following...Ch. 11.1 - What is the output of the following code? public...Ch. 11.1 - Prob. 5STQCh. 11.1 - Complete the definition of the following method....Ch. 11.2 - Revise the method getCount in Listing 11.5 so that...Ch. 11.2 - Prob. 8STQCh. 11.2 - Prob. 9STQCh. 11.2 - Suppose you want me class ArraySearcher to work...
Ch. 11.2 - What Java statement will sort the following array,...Ch. 11.2 - How would you change the class MergeSort so that...Ch. 11.2 - How would you change the class MergeSort so that...Ch. 11.2 - If a value in an array of base type int occurs...Ch. 11.3 - Convert the following event handler to use the...Ch. 11 - What output will be produced by the following...Ch. 11 - What output will be produced by the following...Ch. 11 - Write a recursive method that will compute the...Ch. 11 - Write a recursive method that will compute the sum...Ch. 11 - Complete a recursive definition of the following...Ch. 11 - Write a recursive method that will compute the sum...Ch. 11 - Write a recursive method that will find and return...Ch. 11 - Prob. 8ECh. 11 - Write a recursive method that will compute...Ch. 11 - Suppose we want to compute the amount of money in...Ch. 11 - Prob. 11ECh. 11 - Write a recursive method that will count the...Ch. 11 - Write a recursive method that will remove all the...Ch. 11 - Write a recursive method that will duplicate each...Ch. 11 - Write a recursive method that will reverse the...Ch. 11 - Write a static recursive method that returns the...Ch. 11 - Write a static recursive method that returns the...Ch. 11 - One of the most common examples of recursion is an...Ch. 11 - A common example of a recursive formula is one to...Ch. 11 - A palindrome is a string that reads the same...Ch. 11 - A geometric progression is defined as the product...Ch. 11 - The Fibonacci sequence occurs frequently in nature...Ch. 11 - Prob. 4PPCh. 11 - Once upon a time in a kingdom far away, the king...Ch. 11 - There are n people in a room, where n is an...Ch. 11 - Prob. 7PPCh. 11 - Prob. 10PPCh. 11 - Prob. 12PP
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