Java Recursive - Fibonacci Sequence in Java - JMaster IO

**Here's an explanation of the Fibonacci sequence:**

**What it is:**

- A series of numbers where each number is the sum of the two preceding ones.
- Begins with 0 and 1, followed by 1, 2, 3, 5, 8, 13, 21, 34, ....
- Continues infinitely, following the same rule.

**History:**

- Named after Leonardo of Pisa, known as Fibonacci, who introduced it to Western European mathematics in 1202.
- Appeared earlier in Indian mathematics, as early as 200 BC.

**Mathematical properties:**

- **Golden ratio:** The ratio of successive Fibonacci numbers approaches a constant value of approximately 1.618, known as the golden ratio or phi (Φ). This ratio has unique mathematical properties and is found in various natural phenomena.
- **Fibonacci spiral:** A spiral formed by connecting Fibonacci squares (whose side lengths are Fibonacci numbers) visualizes the golden ratio's relationship to the sequence.
- **Recursion:** The sequence is a classic example of a recursive formula, where each term is defined based on previous terms.

**Appearance in nature:**

- **Plant growth:** The arrangement of leaves on a stem, the branching of trees, and the spiral patterns of pinecones and sunflowers often exhibit Fibonacci numbers.
- **Animal structures:** The spiral shells of snails and the arrangement of scales on pineapples also follow Fibonacci patterns.

**Applications:**

- **Computer science:** Used in algorithms for data compression and sorting.
- **Finance:** Applied in technical analysis to identify potential price retracements and support levels in financial markets.
- **Art and architecture:** Used to create aesthetically pleasing proportions and visual patterns.
- **Music:** Found in musical compositions and the arrangement of notes in some scales.

**Key points:**

- Simple yet profound mathematical sequence with a wide range of applications.
- Reveals patterns in nature, art, and music.
- Demonstrates the beauty and interconnectedness of mathematics.