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Chapter 10: Problem 31

When one examines the units digit of each Fibonacci number \(F_{n}, n \geq 0\), one finds that these digits form a sequence that repeats after 60 terms. [This was first proved by Joseph-Louis Lagrange (1736-1813).] Write a computer program (or develop an algorithm) to calculate this sequence of 60 digits.

Short Answer

Expert verified
The units digits form a repeating sequence of 60 numbers in the Fibonacci sequence. Calculated this sequence using a while loop that appends the units digit of each Fibonacci number into a list. The loop stops when the list reaches length 60, signaling that the repeating sequence of 60 digits has been found.

Step by step solution

01

Understanding Fibonacci Sequence

The Fibonacci sequence is such that each number is the sum of the two preceding ones, starting from 0 and 1. Thus, the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .. and so forth.
02

Initialize Variables and List

To capture the sequence of 60 digits, start by initializing three variables a and b to 0 and 1 respectively (since these are the first two numbers in the Fibonacci sequence), and also intialize an empty list to store the units digits. a and b will be used to calculate the Fibonacci sequence. The list will hold the 60 repeating units digits. This can be done in Python as: 'a, b = 0, 1; units_digits = []'
03

Create Fibonacci Loop and Extract Units Digits

In a while loop, generate the Fibonacci sequence by adding the last two numbers (a + b) and save the result in b while a is updated as the old b. To get the units digit of the Fibonacci numbers, find the modulus of each Fibonacci number by 10 (since any number modulo 10 returns the unit digit of that number). This can be coded as: 'while len(units_digits) < 60: units_digits.append(b % 10); a, b = b, a + b'.
04

Ending the Loop

After each iteration, check if the length of the list has reached 60. If it has, break the loop and print out the sequence of 60 digits. This can be coded as: 'if len(units_digit) == 60: break;. Finally, print out the list of units digits: print(units_digits);'

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Most popular questions from this chapter

For \(n>1\), a permutation \(p_{1}, p_{2}, p_{3}, \ldots, p_{n}\) of the integers \(1,2,3, \ldots, n\) is called orderly if, for each \(i=1,2,3, \ldots\) \(n-1\), there exists a \(j>i\) such that \(\left|p_{J}-p_{l}\right|=1\). [If \(n=2\), the permutations 1,2 and 2,1 are both orderly. When \(n=3\) we find that \(3,1,2\) is an orderly permutation, while \(2,3,1\) is not. (Why not?)] (a) List all the orderly permutations for \(1,2,3\). (b) List all the orderly permutations for \(1,2,3,4\). (c) If \(p_{1}, p_{2}, p_{3}, p_{4}, p_{5}\) is an orderly permutation of \(1,2,3,4,5\), what value(s) can \(p_{1}\) be? (d) For \(n>1\), let \(a_{n}\) count the number of orderly permutations for \(1,2,3, \ldots, n\). Find and solve a recurrence relation for \(a_{n}\).

For \(n \geq 1\), let \(a_{n}\) be the number of ways to write \(n\) as an ordered sum of positive integers, where each summand is at least 2. (For example, \(a_{5}=3\) because here we may represent 5 by 5 , by \(2+3\), and by \(3+2\).) Find and solve a recurrence relation for \(a_{n}\).

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