Some recursive sequences can grow rapidly, others might increase gradually or even decrease at differing speeds. The Fibonacci sequence tends to increase towards infinity, which makes it difficult for us to calculate the nth value in the sequence when n is large. It is believed that “Five hundred years after Fibonacci, Abraham de Moivre developed a theory for solving general linear recurrences and gave the first explicit formula for the values of the Fibonacci sequence: Fn = (φn − (-φ)-n) ⁄ √5, where (1+ √5) ⁄ 2 = φ is the golden ratio” (Huber, p. 29). This formula allows us to solve for any n in the Fibonacci sequence. However, this formula is known as the Binet formula, named after Binet who derived it in 1843 (Weisstein, “Binet’s Fibonacci Number Formula”). Another characteristic of the terms in the Fibonacci sequence is that every third term is even (Paul, p. 1403). This was proven by Neeraj Paul using modular arithmetic; however, this can be proven multiple ways. Essentially, when we add two even values, we will end with an even value. If we add an even to an odd value, then we will end with an odd value. Lastly, when we add two odd values, it results in an even value. Knowing that the Fibonacci sequence always starts with two odd initial terms, we can deduce that all terms except for every third term are not divisible by two, or that they are odd.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| F(n) | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 |
| Odd or Even | Odd | Odd | Even | Odd | Odd | Even | Odd | Odd | Even | Odd |
Figure 3: Every Third Term is Even
By using the Fibonacci sequence and golden ratio, mathematicians have discovered a special spiral known as the golden spiral. It is constructed by building squares with lengths that correspond to the Fibonacci sequence. We start with two squares that share a side, both with sides of length 1. Next, we construct a square that shares a side with our first two squares, making our third square have sides of length 2. By continuing this process, we find that all of the squares have side lengths that follow the Fibonacci sequence. These squares also appear to be rotating and expanding about the first square in a spiral. To construct a spiral, we create an arc between two opposite corners in our first square and then continue the arc from the corner of the next square to the opposite corner. This spiral grows at a rate of 1.618 for each quarter-turn it takes around the center (Katyal, p. 4). That is why it is referred to as the golden spiral. Using the applet below, try to construct the golden spiral.
The golden ratio exhibits some interesting properties that make it such a pleasing number to mathematicians. Consider the following calculations, 1 + 1⁄φ = φ = 1.6180339887 (Kazemi, p. 2). This means that 1⁄φ = φ – 1 = 0.6180339887 is also true. If we were to square φ, we would end up with φ2 = φ + 1 = 2.6180339887. Therefore, φ2 is one more than φ, and φ is one more than 1⁄φ which is a very unique relationship.
Another cool calculation is that the golden ratio when raised to the nth power follows the pattern of the Fibonacci sequence, where φn = Fn * φ + Fn-1 and Fn is the nth Fibonacci number in the sequence (Rollinson). Use the applet below to demonstrate the calculations of this relationship.