Problem 43 - Discrete Mathematics with Applications

Question

Prove each, where \(F_{n}\) is the \(n\) th Fibonacci number, \(L_{n}\) the \(n\) th Lucas number, and \(\alpha=(1+\sqrt{5}) / 2,\) the golden ratio. $$F_{n} \leq 2^{n}, n \geq 1$$

Step-by-Step Solution

Answer

We can prove the inequality \(F_n \leq 2^n\) by mathematical induction. First, verify the base case for \(n=1\), and then assume it holds for \(n=k\). Using the recursive definition of Fibonacci numbers and the properties of Fibonacci numbers, we prove the inequality for \(n=k+1\), and thus conclude that the inequality holds for all \(n \geq 1\).

Step 1: Verify the base case

First, let's check the base case for n = 1. We have the first Fibonacci number to be \(F_1 = 1\), and we need to verify if the inequality \(F_1 \leq 2^1\) holds. \(1 \leq 2\), so the inequality is true for the base case.

Step 2: Basic setup for mathematical induction

Assume that the given inequality \(F_n \leq 2^n\) is true for \(n = k\), where \(k \geq 1\). We need to prove that the inequality is also true for \(n = k + 1\), meaning \(F_{k+1} \leq 2^{k+1}\).

Step 3: Inductive step

Now, by the recursive definition of Fibonacci numbers, we have \(F_{k+1} = F_k + F_{k-1}\). Since we assumed \(F_k \leq 2^k\), and given that the Fibonacci numbers are always positive, it follows that \(F_{k-1} \leq F_k\). Therefore the inequality holds for \(F_{k+1}\) as well: \(F_{k+1} = F_k + F_{k-1} \leq 2^k + 2^k = 2(2^k) = 2^{k+1}\).

Step 4: Conclusion

Since the inequality is true for the base case (n = 1) and we proved the inductive step, it can be concluded that \(F_n \leq 2^n\) for all \(n \geq 1\).

Key Concepts

Fibonacci Numbers

The Fibonacci sequence is a famous series in mathematics, starting with the numbers 0 and 1, where every subsequent number is the sum of the previous two. More formally, the sequence is defined recursively by the equations \(F_0 = 0\), \(F_1 = 1\), and \(F_{n} = F_{n-1} + F_{n-2}\) for \(n \textgreater 1\).

Each number in the sequence is called a Fibonacci number. The sequence appears in many different areas of mathematics and science, often related to growth patterns, spirals, and even financial markets. The concept of Fibonacci numbers is essential to the problem at hand—proving that Fibonacci numbers are bounded by exponential growth, specifically by powers of 2.

Inequality Proof

An inequality proof is a mathematical demonstration that a certain inequality holds true for a given set of elements. In the context of Fibonacci numbers, proving an inequality involves showing that the Fibonacci number \(F_{n}\) is less than or equal to some other value, in this case, \(2^n\), for all values of \(n\) within a particular domain (here, for \(n \textgreater= 1\)).

The act of proving inequalities is crucial in various fields such as optimization, number theory, and algorithm complexity, where understanding the limits and bounds of sequences and functions is necessary.

Inductive Step

The inductive step is a pivotal stage in mathematical induction, a proof technique used to establish the veracity of propositions for all natural numbers. After confirming the base case (the smallest natural number for which the proposition is applicable), the inductive step must show that if the proposition is assumed to be true for some number \(k\), then it must also be true for \(k+1\).

For the Fibonacci number inequality, the proposition involves the Fibonacci number sequence. The crux of the inductive step here is exploiting the recursive nature of the sequence to show that once the inequality holds for a particular \(F_k\), it will also hold for \(F_{k+1}\).

Recursive Definition

Recursive definitions specify a sequence or object in terms of itself. In mathematics, it often means defining a function or sequence using previous terms of the sequence. The Fibonacci sequence is one of the most classic examples of a recursive definition, with each term generated by the sum of the two preceding terms.

This recursive pattern allows mathematicians to apply inductive reasoning when solving problems related to the sequence. The recursive definition plays a fundamental role in the proof for the Fibonacci problem, providing the logical structure needed to apply the inductive step and progressing toward the solution.