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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, Fn = Fn-1 + Fn-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 Fn+1 = 2Fn-1 + Fn-2; and (iv) how this property, Fn+1 = 2Fn-1 + Fn-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

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

Lecturer

University of Oregon