Question

Prove the following statements with either induction, strong induction or proof by smallest counterexample. Concerning the Fibonacci sequence, prove that \(\sum_{k=1}^{n} F_{k}^{2}=F_{n} F_{n+1}\).

Short answer

The equality \(\sum_{k=1}^{n} F_{k}^{2}=F_{n} F_{n+1}\) has been proved true for all positive integers by using mathematical induction involving a base case (n=1), an inductive hypothesis (assuming the equality is true for some arbitrary n), and an inductive step (proving it for \(n+1) assuming it is true for n).

Step-by-step solution

Fibonacci Sequence Definition

The Fibonacci sequence \(F_k\) is a sequence of numbers where each number after the first two is the sum of the two preceding ones. These are typically defined as \(F_1 = 1\), \(F_2 = 1\), and \(F_k = F_{k-1} + F_{k-2}\) for \(k > 2\).

Base Case

For the base case, let's consider n=1. Therefore, the left-hand side (LHS) is \(F_{1}^{2}=1^2=1\). The right-hand side (RHS) is \(F_{1}F_{2} = 1*1=1\). Since LHS = RHS, the proposition holds true for the base case.

Inductive Hypothesis

Assume the proposition is true for some arbitrary positive integer n; that is, \(\sum_{k=1}^{n} F_{k}^{2}=F_{n}F_{n+1}\). This is our inductive hypothesis.

Inductive Step

To prove the proposition for \(n+1\), consider \(\sum_{k=1}^{n+1} F_{k}^{2}= \sum_{k=1}^{n} F_{k}^{2}+F_{n+1}^{2}\). Substituting the inductive hypothesis into the equation gives \(F_{n}F_{n+1} + F_{n+1}^{2}= F_{n+1}(F_{n}+ F_{n+1}) = F_{n+1}F_{n+2}\). This demomonstrates that the equality holds for \(n+1\) when it is true for n.

Key concepts

Mathematical Induction

Mathematical induction is a powerful proof technique used extensively in mathematics to establish the truth of an infinite number of cases. It is akin to a series of dominos falling; if you can prove the first domino falls (the base case) and then show that if any one domino falls, it will cause the next one to fall too (the inductive step), you can conclude all dominoes will fall.

Induction involves two critical steps. First, we verify the base case, ensuring the statement holds for the initial case, often with the smallest value in the domain of natural numbers. The base case acts as the foundation of the proof. Second, assuming that the statement is true for some arbitrary number, termed the inductive hypothesis, we demonstrate it will also hold true for the next number. If these steps are successful, mathematical induction allows us to state the proposition is true for all natural numbers.

The application of this approach can be seen in the exercise provided, where the base case is tested for the first Fibonacci number, and then the property is assumed to hold for an arbitrary Fibonacci number. The proof is completed by showing the property must then also hold for the next number in the sequence.

Strong Induction

Strong induction is a variant of mathematical induction that differs in the inductive step. With strong induction, the assumption is made that not just the previous case but all cases up to that point are true. This means we assume our statement holds for all values from the base case through some integer n, and then use this to prove the statement for n+1.

The key here is that you’re not just relying on the previous ‘domino’, but instead, you can use the entire chain leading up to that point to enforce the next step. This approach can be extremely helpful when the problem at hand is more complex and the proof of any single case might rely on several preceding cases rather than just one.

In the exercise concerning the Fibonacci sequence, if we used strong induction, after establishing the base cases, we would assume the formula holds for all values up to n, rather than just for n, and then proceed to establish it for n+1.

Proof by Smallest Counterexample

Proof by smallest counterexample is a technique where, to prove a statement true, we assume there is a smallest counterexample that disproves it and then demonstrate the existence of such a counterexample leads to a contradiction. Essentially, we leverage the logical principle known as 'reductio ad absurdum', which in Latin means 'reduction to absurdity'.

By showing that if the smallest counterexample were true, it would imply the existence of an even smaller counterexample, which contradicts the assumption that our chosen counterexample is the smallest. As such, we conclude that no such counterexample can actually exist—and hence, the original statement must be true for all cases.

For the Fibonacci sequence problem, employing this technique would involve assuming there is a smallest value of n for which the proposition does not hold, and then using the Fibonacci sequence's definition and properties to show that this leads to a paradox.

Fibonacci Sequence Definition

The Fibonacci sequence is a captivating progression in which each number is the sum of the two preceding ones, generally starting with 0 and 1. It is expressed as: \(F_0 = 0, F_1 = 1,\) and \(F_k = F_{k-1} + F_{k-2}\) for \(k > 1\).

This sequence presents itself in various aspects of art, nature, and science. For instance, the spiral patterns observed in pinecones and sunflowers can be described using Fibonacci numbers. Each term in the sequence is called a Fibonacci number and the sequence exhibits remarkable properties, one of which is highlighted in the exercise provided. That property points out an interesting relationship between the sum of the squares of these numbers and the product of a number in the sequence with its subsequent number, enhancing the beauty and harmony found within the Fibonacci sequence.