Question

How many "words" of \(k\) letters can be made from the letters \(\\{a, b\\}\) if there are no adjacent \(a\) 's?

Short answer

The number of such words is given by the \((k+2)\)th Fibonacci number.

Step-by-step solution

Understand the Problem

You need to count how many sequences ("words") of length \(k\) can be formed using the letters \(\{a, b\}\) such that no two \('a'\) letters are next to each other.

Define the Base Cases

For \(k = 1\), the possible words are \('a'\) and \('b'\), yielding 2 words in total.

Use Recursion for Larger k

We need a recursive way to build solutions. Let \(a_k\) be the count of valid words of length \(k\). We'll split this into two parts: ending with 'a' and ending with 'b'. If a word of length \(k\) ends with 'a', then the previous letter must be 'b', making it \(b_{k-1}\). If it ends with 'b', it could be either a valid word of length \(k-1\) followed by 'b', i.e., \(a_{k-1}\). So, the recurrence relation is \(a_k = a_{k-1} + b_{k-1}\) and \(b_k = a_k\).

Simplify the Recurrence Relation

Recognizing that \(b_k = a_k\), we simplify it to: \(a_k = a_{k-1} + a_{k-2}\). This is the Fibonacci sequence, where \(F_1 = 2\) and \(F_2 = 3\) to start.

Calculate Through Iteration

Consider the base case for \(k=1\): \(a_1 = 2\). For \(k=2\): \(a_2 = 3\). Further calculations follow the Fibonacci sequence incrementing from these initial conditions. For example, \(a_3 = a_2 + a_1 = 3 + 2 = 5\). Continue this for larger \(k\).

Key concepts

Fibonacci sequence

The Fibonacci sequence is a famous number sequence where each number is the sum of the two preceding ones, typically starting with 0 and 1. In mathematical terms, it's expressed as:

• \( F_0 = 0 \)
• \( F_1 = 1 \)
• \( F_n = F_{n-1} + F_{n-2} \text{ for } n > 1 \)

This sequence naturally appears in various domains, such as mathematics, computer science, and nature, and is characterized by its incremental pattern. In our exercise, we see a slight variation because it starts with different base cases, but the recursive structure follows the same pattern. This adaptation maintains the essence of the Fibonacci series, serving specific constraints, like ensuring no two 'a's are adjacent. Understanding this sequence helps us structure complex problems into simpler recursive tasks that mirror real-world relationships.

Recurrence relations

Recurrence relations are equations that recursively define sequences. By describing each term as a function of the preceding ones, they help solve complex combinatorial problems succinctly. They are foundational in computer science, especially for algorithms involving iterative calculations, like sorting algorithms or dynamic programming.In our exercise, the problem involves finding a relation that defines the number of valid words, given the constraints. We derived:

• \( a_k = a_{k-1} + a_{k-2} \)

Here, each term in the sequence depends on the previous two values. This relation mirrors the Fibonacci sequence, making it easy to leverage well-known properties and solutions from Fibonacci when calculating the number of words for larger \(k\). Recurrence relations guide us in breaking down complex sequences into more manageable calculations, turning a cumbersome problem into an easier one through recursive breakdown.

Base cases

Base cases are crucial in recursive problems. They define the simplest instances, serving as the foundation for building more complex solutions. Without them, a recursive approach lacks a starting point, which can result in endless calculations.In our exercise, the base cases are:

• For \( k = 1 \), there are two words: 'a' and 'b'. Hence, \( a_1 = 2 \).
• For \( k = 2 \), the valid words are 'ab' and 'ba'. Thus, \( a_2 = 3 \).

These cases provide the necessary anchor for the recurrence. They help demonstrate how simple solutions lead to the larger recursive construction. As we calculate for larger \( k \), these base cases become the initial conditions, ensuring the accuracy of the recursive models. Base cases, therefore, not only simplify recursive logic but also guarantee the feasibility of recursive strategies in finding the solution.