Fibonacci numbers

Definition

Fibonacci numbers form a sequence where each number is the sum of the two preceding ones, starting from 0 and 1. The general formula for the nth Fibonacci number is $F_n = F_{n-1} + F_{n-2}$.

5 Must Know Facts For Your Next Test

1. The first two Fibonacci numbers are defined as $F_0 = 0$ and $F_1 = 1$.
2. The sequence progresses as: 0, 1, 1, 2, 3, 5, 8, etc.
3. Fibonacci numbers can be represented using Binet's formula: $F_n = \frac{\phi^n - (1-\phi)^n}{\sqrt{5}}$ where $\phi$ (the golden ratio) is approximately equal to 1.618.
4. They exhibit exponential growth proportional to the golden ratio.
5. Fibonacci numbers appear in various natural phenomena and are used in algorithm design and analysis.

Review Questions

• What is the recursive formula for generating Fibonacci numbers?
• How are the first two Fibonacci numbers defined?
• Explain how Binet's formula relates to Fibonacci numbers.

Related terms

Golden Ratio: An irrational number approximately equal to $1.618$, often denoted by $\phi$, which appears frequently in mathematics and nature.

Recursive Sequence: A sequence in which each term is defined as a function of one or more of its preceding terms.

Binet's Formula: $F_n = \frac{\phi^n - (1-\phi)^n}{\sqrt{5}}$, an explicit expression for finding the nth Fibonacci number using powers of the golden ratio.