Problem 7 - Discrete Mathematics with Applications

Question

The \(n\) th Lucas number \(L_{n},\) named after the French mathematician François-Edouard-Anatole Lucas, is defined recursively as follows: $$\begin{array}{l} L_{1}=1, \quad L_{2}=3 \\ L_{n}=L_{n-1}+L_{n-2}, n \geq 3 \end{array}$$

Step-by-Step Solution

Answer

To find the nth Lucas number, use the base cases: \(L_1 = 1\) and \(L_2 = 3\), and the recursive relation: \(L_n = L_{n-1} + L_{n-2}\) for \(n \ge 3\). Start from \(n=3\) and calculate the Lucas numbers step by step, using the recursive formula, until the desired value of n is reached.

Step 1: Understand the Base Cases

First, we need to know the base cases. They are given as: \(L_1 = 1\) and \(L_2 = 3\) These are the first two starting points of the Lucas sequence.

Step 2: Understand the Recursion

We are given the recursive relation: \(L_n = L_{n-1} + L_{n-2}\), for \(n \ge 3\) This tells us that to find the nth term, we need to add the two terms before the nth term. This formula holds for \(n \ge 3\), as we already have the values for \(n=1\) and \(n=2\).

Step 3: Calculate the nth Lucas number

Using the recursive relation, we will calculate the nth Lucas number step by step. We will start from \(n=3\) and move forward to find the desired Lucas number. 1. For n = 3, using the recursive formula: \(L_3 = L_{3-1} + L_{3-2}\) \(L_3 = L_{2} + L_{1}\) \(L_3 = 3 + 1 = 4\) 2. For n = 4, using the recursive formula: \(L_4 = L_{4-1} + L_{4-2}\) \(L_4 = L_{3} + L_{2}\) \(L_4 = 4 + 3 = 7\) 3. Repeat this process until you reach the desired value of n. We can either solve it by hand using the steps shown above, or use a programming algorithm or a spreadsheet to calculate it for larger values of n.

Key Concepts

Recursive Sequences

Recursive sequences are defined using operations on one or more of the preceding terms to generate each subsequent term. In the case of Lucas numbers, the sequence is defined with the formula:\[ L_n = L_{n-1} + L_{n-2} \]where each term is the sum of the two preceding terms. This allows the sequence to grow based on its own historical values.

• The sequence begins with predefined base cases, which we'll discuss next.
• Each additional term is calculated using the recursive formula, essentially building upon what has come before.

Recursive sequences are not limited to simple addition; they can include any number of operations or rules as long as each term is created from previous ones.
Understanding recursive sequences offers insight into how patterns evolve over iterations, which can be widely useful in computer science, mathematics, and various computational problems.

Base Cases

Base cases are critical in recursive definitions. They serve as the starting point for the sequence. For Lucas numbers, these cases are:

• \( L_1 = 1 \)
• \( L_2 = 3 \)

These initial terms are explicitly given and do not require a formula to determine. They allow the recursive formula to function effectively by providing the necessary starting values.

Importance of Base Cases

The base cases ensure that the sequence can begin generating subsequent terms using its rule. Without these, the formula has no initial values to act upon. Base cases also ensure that recursion does not enter an infinite loop by providing a known state from which to proceed.
In mathematics, base cases are part of a hypothesis's foundation, allowing researchers to prove broader properties about sequences using other techniques like mathematical induction.

Mathematical Induction

Mathematical induction is a technique for proving statements about integers. It is particularly powerful for validating properties of recursively defined sequences like the Lucas numbers. The process of mathematical induction involves two main steps:

• Base case: Prove that the statement holds true for the initial value(s). For Lucas numbers, verify the base cases: \( L_1 = 1 \) and \( L_2 = 3 \).
• Inductive step: Show that if the statement holds for an arbitrary integer \( k \), then it must also hold for \( k+1 \). For recursive sequences, assume \( L_k = L_{k-1} + L_{k-2} \) holds and prove that \( L_{k+1} = L_k + L_{k-1} \).

By confirming these steps, you establish that a property is true for all integers beyond the base case.

Using Mathematical Induction

In practice, mathematical induction can validate recurring relationships, similar to those found in Lucas numbers. It is essential in analyzing types of sequences where the behavior is defined recursively, ensuring their general statements and formulae are universally applicable.