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# Direct Proofs Using Fibonacci Numbers

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Proofs: Number Theory and Sequences

In this course, Professor Shabnam Akhtari (University of Oregon) explores proofs, specifically those motivated by number theory and sequences. In the first mini-lecture, we introduce the Fibonacci numbers and use them to motivate examples of direct proofs. In the second mini-lecture, we introduce proof by contradiction and use this method to prove properties about prime numbers, including Euclid’s Theorem, which states that there are infinitely many prime numbers. In the third mini-lecture, we discuss prime factorisation notation using exponents and work through two related proofs. In the fourth mini-lecture, we consider direct proofs involving arithmetic and geometric sequences.

Direct Proofs Using Fibonacci Numbers

In this mini-lecture, we use the Fibonacci numbers to motivate examples of direct proofs. As we move though this mini-lecture, we consider: (i) the definition of Fibonacci numbers, F_{n} = F_{n-1} + F_{n-2}, which can be used to create a sequence of numbers; (ii) mathematical patterns in the Fibonacci sequence, such as every third number being even while all others are odd; (iii) a direct proof of a property of Fibonacci numbers, which states that F_{n+1} = 2F_{n-1} + F_{n-2}; and (iv) how this property, F_{n+1} = 2F_{n-1} + F_{n-2}, can also help us deduce which terms are even and which are odd in the Fibonacci sequence.

#### Cite this Lecture

**APA style**

Akhtari,
S.
(2022, August 30).
*Proofs: Number Theory and Sequences - Direct Proofs Using Fibonacci Numbers* [Video]. MASSOLIT. https://massolit.io/courses/proofs-number-theory-and-sequences/direct-proofs-using-fibonacci-numbers

**MLA style**

Akhtari,
S.
"Proofs: Number Theory and Sequences – Direct Proofs Using Fibonacci Numbers." *MASSOLIT*, uploaded by MASSOLIT,
30 Aug 2022,
https://massolit.io/courses/proofs-number-theory-and-sequences/direct-proofs-using-fibonacci-numbers