## Main Concepts

Binet’s Formula Proof, Fibonacci sequence

Vector space, Linear space

Dimension and basis of vector space

Linear combinations

## Duration

60 minutes in class

## Summary of Activity

In this in-depth discovery of some special features of the Fibonacci sequence, we introduce, use and prove Binet’s formula using linear algebra. Additionally, we present several striking extensions of the Fibonacci sequence that almost no student knows about in spite of the popularity of the sequence in high school math courses from Geometry to Precalculus.

## Notes and Insights

Questions (d) through (i) are mostly guided by the teacher, but the rest should be done independently by students. Fun facts about Fibonacci sequences (source: The Man Who Liked Only Numbers)
1. The Fibonacci sequence includes primes (2, 3, 5, 13, etc.). Nobody knows if there’s an infinite number of them.
2. Are any Fibonacci-like sequences prime-free? Yes, obviously you can start with (2, 4) and all terms will be multiples of 2… But can we make a Fibonacci-like sequence that is prime-free with two relatively-prime numbers? The smallest known starting points have sixteen and seventeen digits: (3794765361567513, 20615674205555510) !!!