Question

Prove the following statements with either induction, strong induction or proof by smallest counterexample. Concerning the Fibonacci sequence, prove that \(F_{1}+F_{2}+F_{3}+F_{4}+\cdots+F_{n}=F_{n+2}-1\).

Short answer

The property \(F_{1}+F_{2}+F_{3}+F_{4}+...+F_{n}=F_{n+2}-1\) of the Fibonacci sequence is proved to be true for all natural numbers \(n\) by using mathematical induction, where we verified the base cases and showed that if the statement holds for an arbitrary integer \(n\), it also holds for \(n+1\).

Step-by-step solution

Understanding Fibonacci Sequence & Proving Base Cases

The Fibonacci sequence is defined such that \(F_{1}=1\), \(F_{2}=1\), and for all \(n>2\), \(F_{n}=F_{n-1}+F_{n-2}\). Now, to use induction, we need to verify the base cases. For \(n=1\), we have \(F_{1}=1\) on the left hand side, and on the right hand side we have \(F_{1+2}-1=F_{3}-1=2-1=1\). Hence, the proposition holds true for \(n=1\). Similarly, For \(n=2\), we have \(F_{1}+F_{2}=1+1=2\) on the left hand side, and on the right hand side we have \(F_{2+2}-1=F_{4}-1=3-1=2\). Hence, the proposition holds true for \(n=2\) as well.

Inductive Step

Now, let's assume that the proposition holds for some arbitrary integer \(n\), i.e., \(F_{1}+F_{2}+F_{3}+F_{4}+\cdots+F_{n}=F_{n+2}-1\). We need to prove that the proposition holds for \(n+1\), i.e., \(F_{1}+F_{2}+F_{3}+F_{4}+\cdots+F_{n}+F_{n+1}=F_{n+3}-1\). To find \(F_{n+3}\), we use the relationship \(F_{n+3}=F_{n+2}+F_{n+1}\). Substituting our assumption into this, we get \(F_{n+3}=F_{n+2}+F_{n+1}=(F_{1}+F_{2}+F_{3}+F_{4}+\cdots+F_{n} +1)+F_{n+1}\), or \( F_{n+3}-1=(F_{1}+F_{2}+\cdots+F_{n})+F_{n+1}\), which is what we wanted to prove. Thus, the statement holds for \(n+1\) given it holds for \(n\).

Inference

Since the base cases hold and the statement for \(n+1\) holds given that the statement for \(n\) holds, we can conclude from the principle of mathematical induction that the given equation \(F_{1}+F_{2}+F_{3}+F_{4}+\cdots+F_{n}=F_{n+2}-1\) holds for all \(n\) that are natural numbers

Key concepts

Fibonacci sequence

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, typically starting with 0 and 1. In the case of the mentioned exercise, the sequence starts with 1 and 1. That is, for any positive integer index n, the Fibonacci number at that position, denoted as F_n, is defined as F_n = F_{n-1} + F_{n-2}, with seed values F_1 = 1 and F_2 = 1.

The Fibonacci sequence beautifully illustrates how each term contributes to the creation of subsequent terms and finds applications in fields ranging from computer algorithms to biological phenomena. Understanding this principle is fundamental when exploring mathematical patterns and helps students to intuitively grasp progression and growth within numerical sequences.

Proof by induction

Proof by induction is a mathematical technique used to establish that a statement is true for all natural numbers. It consists of two crucial steps: the base case and the inductive step.

To apply proof by induction:

• Base Case: Verify the truth of the statement for the initial value, usually n=1.
• Inductive Step: Assume the statement holds for an arbitrary natural number n, and then prove that the statement must also hold for n+1, based on this assumption.

If both of these steps are successfully shown, the principle of mathematical induction allows us to conclude the statement is true for all natural numbers. This proof method is incredibly powerful in mathematics as it gives us a solid foundation to prove propositions that recur over an infinite set of elements.

Proof techniques

When tackling mathematical statements, various proof techniques can be employed depending on the nature of the problem. Direct proofs work by straightforwardly showing the truth of a statement from axioms and previously established theorems. In contrast, contradiction and contrapositive are indirect methods, where the truth of a statement is demonstrated by showing the falsehood of its negation or the falsehood of the reverse implication, respectively.

Proof by smallest counterexample involves assuming that the statement is false and then demonstrating a contradiction by identifying the so-called 'smallest counterexample' to the statement, ultimately showing that such a counterexample cannot exist. Each technique is a tool in a mathematician's toolkit, chosen based on the proposition and overall strategy for demonstrating the veracity of mathematical truth.

Inductive step

The inductive step is a pivotal component of proof by induction and refers to the process where we assume the proposition holds for an arbitrary natural number n (the induction hypothesis) and then prove it holds for the subsequent number n+1. This process includes strategically utilizing the assumption to bridge n to n+1.

In the case of the Fibonacci sequence problem, the inductive step involves showing that if the sum of the first n Fibonacci numbers equals F_{n+2} - 1, then the sum up to n+1 terms is F_{n+3} - 1. By confirming the statement for n+1, using the assumption for n, we are able to 'climb the ladder' of natural numbers, thereby proving the relationship holds for the entire sequence. This elegant leap from 'n' to 'n+1' exemplifies the beauty and logical power of inductive reasoning.