Question

Prove that the number of \(n\) -digit binary numbers that have no consecutive 1's is the Fibonacci number \(F_{n+2}\). For example, for \(n=2\) there are three such numbers \((00,01,\) and 10\()\), and \(3=F_{2+2}=F_{4}\). Also, for \(n=3\) there are five such numbers \((000,001,010,100,101),\) and \(5=F_{3+2}=F_{5}\).

Short answer

The total count of \(n\)-digit binary numbers that have no consecutive 1's is represented by the Fibonacci number \(F_{n+2}\). This is proven by partitioning the \(n\)-digit binary numbers into two sets based on their starting digit (either '0' or '1'), each of which corresponds to a Fibonacci number.

Step-by-step solution

Understand the Fibonacci Sequence

The Fibonacci sequence is a series of numbers where a number is found by adding up the two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth. Written mathematically, it is: \(F_{0} = 0, F_{1} = 1, F_{i} = F_{i-1} + F_{i-2}\) for \(i \ge 2\)

Binary Representation without Consecutive 1's

Consider \(n\) digit binary numbers. They can be divided into two sets: the first set starting with '0' and the second set starting with '1'. Since the first digit is decided, for the remaining \(n-1\) digits, there can be \(F_{n+1}\) binary numbers without consecutive 1's. The second set cannot have '1' as the next digit, therefore there are \(F_{n}\) binary numbers of length \(n-1\). Thus, total number is \(F_{n+1} + F_{n} = F_{n+2}\).

Overview of the Proof

Based on the steps above, it can be observed that if you list out all possible \(n\)-digits binary numbers that start with '0' and '1', this partitions all \(n\)-digits binary numbers into two sets without having any overlap. This clearly indicates that total count will be \(F_{n+2}\), which is a Fibonacci relation. The recurrence relation agrees with that of Fibonacci, thus completing the proof by the principle of mathematical induction.

Key concepts

Mathematical Induction

Mathematical induction is a proof technique that is used to establish the truth of an infinite number of cases by proving a base case and a step case.
Imagine you have a long row of dominoes. If you can show the first one falls (the base case) and then, if any one domino falls, the next one will also fall (the induction step), you can conclude all the dominos will fall.
In the context of Fibonacci numbers in binary sequences, we use induction to show that a property holds for the base case, such as the number of 2 or 3-digit binary sequences without consecutive 1's. Then, we prove that if it holds for an arbitrary number of digits, it also holds for one more digit. This process firmly establishes that the property will be true for all lengths of binary sequences, just as we can trust all the dominoes will fall in succession.

Binary Numbers

Binary numbers are the heart of computer systems and digital electronics. They are numbers composed only of the digits 0 and 1 and are the foundation of all binary code, which is used to write data such as computer processor instructions or digital text.
In the given exercise, our focus is on binary numbers without consecutive 1's because such sequences have interesting properties related to the Fibonacci sequence. We observe patterns where these sequences are enumerated based on their length, leading to intriguing combinatorial relationships.

Fibonacci Sequence

The Fibonacci sequence is a famous series in numbers that has fascinated mathematicians, scientists, and artists alike. It begins with 0 and 1, and each subsequent number is the sum of the two preceding ones. It has countless applications and appearances in nature, like the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, and much more.
In our problem, we see this pattern emerging in the context of binary sequences without consecutive 1's. The connection may not be immediately apparent, but through exploration and application of mathematical induction, we uncover that the two are indeed intricately linked.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting, combination, and permutation of sets. It's fundamental to various fields including computer science, probability, and algebra.
In the context of this problem, combinatorics allow us to count the number of binary sequences of a certain length without consecutive 1's. Recognizing the combinatorial structure of these sequences enables us to form a neat parallel to the Fibonacci sequence, with each term expressing a count of a particular combinatorial set—that set being the binary numbers without consecutive ones corresponding to that length.